Rear projection optical system

ABSTRACT

An oblique projection optical system for leading rays of light from a display surface on which an image is displayed to a projection surface in such a way that the ray of light from the center of the display surface is obliquely incident on the projection surface in order to project a magnified image of the image displayed on the display surface onto the projection surface includes a plurality of reflecting surfaces having a power. At least two of the reflecting surfaces have a free-form curved surface, and, of all the reflecting surfaces, the one closest to the projection surface has a negative power and at least one of the other has a positive power. Alternatively, in a rear projection optical system having a projection optical system for projecting an image displayed on a panel display surface onto a screen surface, the projection optical system includes at least four curved-surface reflecting mirrors.

[0001] This application is based on Japanese Patent Applications Nos.2001-363653 and 2000-34319 filed on Nov. 29, 2001 and Feb. 7, 2000respectively, the contents of which are hereby incorporated byreference.

BACKGROUND OF THE INVENTION

[0002] 1. Field of the Invention

[0003] The present invention relates to a projection optical system forprojecting a magnified image on a screen, and to a rear projectionoptical system provided with such a projection optical system. Morespecifically, the present invention relates to an oblique projectionoptical system and a rear projection optical system that shine a beam oflight on a screen from an oblique direction.

[0004] 2. Description of the Prior Art

[0005] From long ago, it has been common practice to project a magnifiedimage of an image displayed on a small display surface onto a screen.Before, projection of an image on a screen was generally achievedthrough front projection, whereby the image is projected from in frontof the screen, i.e. from the same side as the observer, for example aswhen a movie is shown in a movie theater. These days, projection of animage is achieved also through rear projection, whereby the image isprojected from behind the screen by the use of a screen that transmitslight. Today, large-screen television sets adopting rear projection arein practical use.

[0006] It is desired that, except in cases where a large facility like amovie theater itself constitutes a projection apparatus, a projectionapparatus be provided with a large screen and be simultaneously compact.In particular, in a rear projection apparatus that projects an imagefrom behind a screen, it is desired that the apparatus be slim, i.e.that its depth dimension in the direction perpendicular to the screen besmall.

[0007] In early models of rear projection apparatus, to make them slim,a very common centered optical system is used as a projection opticalsystem, and a flat-surface mirror is arranged behind a screen so as toturn the optical path of the light exiting from the powered part of theprojection optical system. However, to prevent distortion in the imageformed on the screen, the optical path along the optical axis turned bythe flat-surface mirror needs to run through the center of the screenperpendicularly thereto. This makes it difficult to slim down theapparatus below a certain thickness. The optical path is turnedvertically, because then the turned optical path is shorter than if itis turned horizontally, and usually all the parts, including the displaysurface on which an image is displayed, other than the flat-surfacemirror for turning the optical path are arranged below the screen.

[0008] An effective way to further slim down rear projection apparatusis to adopt oblique projection, in which the ray of light striking thecenter of the screen, i.e. the ray representing the center of the image,is incident on the screen at a large angle of incidence. However,attempting to achieve oblique projection with a centered projectionoptical system necessitates making the optical path along the opticalaxis turned by the flat-surface mirror run off the center of the screen.Accordingly, the projection optical system needs to include alarge-diameter wide-angle lens of which only part is used forprojection. Such an optical system can be realized, but it entails highcost, and in addition makes the projection optical system itself larger,with little effect of slimming down the apparatus.

[0009] To overcome this, proposals have been made to use reflectingmirrors with curved surfaces as powered elements included in theprojection optical system. For example, Re-published Patent ApplicationNo. WO 97/01987 proposes a projection optical system composed of fourcurved-surface mirrors. These curved-surface mirrors have, in order fromthe display surface side, a positive, a negative, a positive, and anegative power. The curved surface closest to the display surface is aspherical surface, and the other three curved surfaces are asphericalsurfaces. The projection optical system that the applicant of thepresent invention proposes in Japanese Patent Application Laid-Open No.2001-221949 also is composed of four curved-surface mirrors. In thisprojection optical system, the curved-surface mirrors have, in orderfrom the display surface side, a positive, a positive, a negative, and anegative power, or a positive, a positive, a negative, and a positivepower. All these surfaces are spherical or aspherical surfaces. Inaddition to these publications, there are more that propose projectionoptical systems composed of three curved-surface mirrors and of othertypes.

[0010] Conventionally, an oblique projection optical system composed ofcurved-surface mirrors is, to minimize the lowering of imagingperformance, designed to have a large f-number, and has a long opticalpath length from the display surface on which an image is displayed tothe projection surface at which a screen is arranged. Moreover, to slimdown the apparatus incorporating it while securing a long optical pathlength, its optical path is turned many times with flat-surface mirrors.The optical path needs to be turned, except on the last occasion, aroundthe screen, specifically below or above the screen, so as not to hinderthe projection of the image on the screen. Thus, an oblique projectionoptical system composed of curved-surface mirrors helps slim down theapparatus incorporating it, but does not contribute to reducing theheight dimension thereof. Moreover, in a conventional oblique projectionoptical system, only necessary parts of curved-surface mirrors are usedso as not to hinder miniaturization. Anyway, all these mirrors havespherical or aspherical surfaces that are symmetric about an axis.

[0011] As long as a long optical path length is secured to prevent thelowering of imaging performance, it is difficult to reduce the heightdimension of the screen without sacrificing the flatness of theapparatus. Thus, modern oblique projection optical systems areconsidered to have almost reached the limit in terms of the trade-offbetween the slimming-down of projection apparatus and the reduction ofthe height dimension thereof.

[0012] On the other hand, as described above, rear projection opticalsystems used in common rear projection apparatus achieve theslimming-down of the apparatus by turning the optical path of the lightexiting from a projection optical system with a single reflecting mirrorarranged behind a screen. However, the projection optical system usedhere is of a centered type, and therefore the ray striking the center ofthe screen surface needs to be substantially perpendicular to the screensurface. This makes it difficult to slim down rear projection opticalsystems below a certain thickness.

[0013] To overcome this, various optical arrangements have been proposedfor further slimming-down. For example, Japanese Patent Registered No.2932609 and Japanese Patent Applications Laid-Open No. H3-87731,H2-153338, H2-146535, and H2-130543 disclose rear projection opticalsystems in which the optical path of a projection optical system isturned with two flat-surface reflecting mirrors.

[0014] However, with conventional rear projection optical systems,sufficient slimming-down is difficult, or slimming them down poses newproblems. For example, the rear projection optical system disclosed inJapanese Patent Registered No. 2932609 mentioned above adopts a methodusing a re-imaging projection optical system in which a displayed imageis first imaged, and the resulting image is then projected on a screensurface so as to be imaged again. This inevitably makes the projectionoptical system large. In addition, this method requires an obliqueprojection optical system that permits the ray striking the center ofthe screen surface to be incident thereon at a large angle of incidence,but the publication describes no specific optical arrangement of such anoptical system. The rear projection optical systems disclosed inJapanese Patent Applications Laid-Open No. H3-87731, H2-153338,H2-146535, and H2-130543 mentioned above also require oblique projectionoptical systems for slimming-down, but the publications do not make itclear what specific optical arrangement to use as a projection opticalsystem.

[0015] An oblique projection optical system is usually realized by usingpart of a centered optical system. However, to slim down a rearprojection optical system, the projection angle of the principal rayneeds to be made very large. Thus, it is inevitable to use part of avery wide-angle centered optical system. In general, a wide-angleoptical system requires a large number of lens elements, and their lensdiameters are very large. This makes the optical system as a wholelarge.

[0016] Some display apparatus incorporate a rear projection opticalsystem that is slimmed down by actually adopting an oblique projectionoptical system employing curved-surface reflecting mirrors. In thesedisplay apparatus, however, the light that has exited from theprojection optical system is reflected directly by a flat-surfacereflecting mirror arranged behind a screen, and thus the curved-surfacereflecting mirror that constitutes the last surface of the projectionoptical system needs to be very large. A large curved-surface reflectingmirror like this is disadvantageous in terms of mass production andcost. Moreover, if the projection optical system includes only threecurved-surface mirrors, it is highly sensitive to errors, and is thusdifficult to manufacture.

SUMMARY OF THE INVENTION

[0017] An object of the present invention is to provide an obliqueprojection optical system that offers high imaging performance and thatpermits further miniaturization not only in the direction perpendicularto the screen but also in the direction along the screen.

[0018] Another object of the present invention is to provide a rearprojection optical system that offers satisfactory optical performancebut is nevertheless advantageous in terms of mass production and costand that is slim and is composed of compact optical components.

[0019] To achieve the above objects, according to one aspect of thepresent invention, an oblique projection optical system for leading raysof light from a display surface on which an image is displayed to aprojection surface in such a way that the ray of light from the centerof the display surface is obliquely incident on the projection surfacein order to project a magnified image of the image displayed on thedisplay surface onto the projection surface includes a plurality ofreflecting surfaces having a power. Here, at least two of the reflectingsurfaces having a power have a free-form curved surface, and, of all thereflecting surfaces having a power, the one closest to the projectionsurface has a negative power and at least one of the other has apositive power.

[0020] According to another aspect of the present invention, in a rearprojection optical system having a projection optical system forprojecting an image displayed on a panel display surface onto a screensurface, the projection optical system includes at least fourcurved-surface reflecting mirrors.

BRIEF DESCRIPTION OF THE DRAWINGS

[0021] This and other objects and features of the present invention willbecome clear from the following description, taken in conjunction withthe preferred embodiments with reference to the accompanying drawings inwhich:

[0022]FIG. 1 is a sectional view, taken along the x-y plane, of theprojection optical system of a first embodiment of the invention;

[0023]FIG. 2 is a side view, as seen from the z direction, of theprojection optical system of the first embodiment;

[0024]FIG. 3 is a top view, as seen from the y direction, of theprojection optical system of the first embodiment;

[0025]FIG. 4 is a front view, as seen from the x direction, of theprojection optical system of the first embodiment;

[0026]FIG. 5 is a spot diagram obtained on the projection surface of theprojection optical system of the first embodiment;

[0027]FIG. 6 is a diagram showing the distortion observed on theprojection surface of the projection optical system of the firstembodiment;

[0028]FIG. 7 is a sectional view, taken along the x-y plane, of theprojection optical system of a modified example of the first embodiment;

[0029]FIG. 8 is a top view, as seen from the y direction, of theprojection optical system of the modified example of the firstembodiment;

[0030]FIG. 9 is a sectional view, taken along the x-y plane, of theprojection optical system of another modified example of the firstembodiment;

[0031]FIG. 10 is a sectional view, taken along the x-y plane, of theprojection optical system of a second embodiment of the invention;

[0032]FIG. 11 is a side view, as seen from the z direction, of theprojection optical system of the second embodiment;

[0033]FIG. 12 is a top view, as seen from the y direction, of theprojection optical system of the second embodiment;

[0034]FIG. 13 is a spot diagram obtained on the projection surface ofthe projection optical system of the second embodiment;

[0035]FIG. 14 is a diagram showing the distortion observed on theprojection surface of the projection optical system of the secondembodiment;

[0036]FIG. 15 is a sectional view, taken along the x-y plane, of theprojection optical system of a third embodiment of the invention;

[0037]FIG. 16 is a side view, as seen from the z direction, of theprojection optical system of the third embodiment;

[0038]FIG. 17 is a top view, as seen from the y direction, of theprojection optical system of the third embodiment;

[0039]FIG. 18 is a spot diagram obtained on the projection surface ofthe projection optical system of the third embodiment;

[0040]FIG. 19 is a diagram showing the distortion observed on theprojection surface of the projection optical system of the thirdembodiment;

[0041]FIG. 20 is a sectional view, taken along the x-y plane, of theprojection optical system of a fourth embodiment of the invention;

[0042]FIG. 21 is a side view, as seen from the z direction, of theprojection optical system of the fourth embodiment;

[0043]FIG. 22 is a top view, as seen from the y direction, of theprojection optical system of the fourth embodiment;

[0044]FIG. 23 is a spot diagram obtained on the projection surface ofthe projection optical system of the fourth embodiment;

[0045]FIG. 24 is a diagram showing the distortion observed on theprojection surface of the projection optical system of the fourthembodiment;

[0046]FIG. 25 is a sectional view, taken along the x-y plane, of theprojection optical system of a fifth embodiment of the invention;

[0047]FIG. 26 is a side view, as seen from the z direction, of theprojection optical system of the fifth embodiment;

[0048]FIG. 27 is a top view, as seen from the y direction, of theprojection optical system of the fifth embodiment;

[0049]FIG. 28 is a spot diagram obtained on the projection surface ofthe projection optical system of the fifth embodiment;

[0050]FIG. 29 is a diagram showing the distortion observed on theprojection surface of the projection optical system of the fifthembodiment;

[0051]FIG. 30 is a sectional view, taken along the x-y plane, of theprojection optical system of a sixth embodiment of the invention;

[0052]FIG. 31 is a side view, as seen from the z direction, of theprojection optical system of the sixth embodiment;

[0053]FIG. 32 is a top view, as seen from the y direction, of theprojection optical system of the sixth embodiment;

[0054]FIG. 33 is a spot diagram obtained on the projection surface ofthe projection optical system of the sixth embodiment;

[0055]FIG. 34 is a diagram showing the distortion observed on theprojection surface of the projection optical system of the sixthembodiment;

[0056]FIG. 35 is a sectional view, taken along the x-y plane, of theprojection optical system of a seventh embodiment of the invention;

[0057]FIG. 36 is a side view, as seen from the z direction, of theprojection optical system of the seventh embodiment;

[0058]FIG. 37 is a top view, as seen from the y direction, of theprojection optical system of the seventh embodiment;

[0059]FIG. 38 is a spot diagram obtained on the projection surface ofthe projection optical system of the seventh embodiment;

[0060]FIG. 39 is a diagram showing the distortion observed on theprojection surface of the projection optical system of the seventhembodiment;

[0061]FIG. 40 is an optical path diagram of the rear projection opticalsystem of an eighth embodiment of the invention;

[0062]FIG. 41 is a diagram showing the projection optical systemconstituting the eighth embodiment and a principal portion of theoptical path thereof;

[0063]FIG. 42 is a spot diagram of the eighth embodiment;

[0064]FIG. 43 is a distortion diagram of the eighth embodiment;

[0065]FIG. 44 is an optical path diagram of the rear projection opticalsystem of a ninth embodiment of the invention;

[0066]FIG. 45 is a diagram showing the projection optical systemconstituting the ninth embodiment and a principal portion of the opticalpath thereof;

[0067]FIG. 46 is a spot diagram of the ninth embodiment;

[0068]FIG. 47 is a distortion diagram of the ninth embodiment;

[0069]FIG. 48 is an optical path diagram of the rear projection opticalsystem of a tenth embodiment of the invention;

[0070]FIG. 49 is a diagram showing the projection optical systemconstituting the tenth embodiment and a principal portion of the opticalpath thereof;

[0071]FIG. 50 is a spot diagram of the tenth embodiment;

[0072]FIG. 51 is a distortion diagram of the tenth embodiment; and

[0073]FIG. 52 is a diagram showing the structure of a principal portionof a screen suitable for use in the eighth to tenth embodiments and theoptical path therethrough.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0074] Hereinafter, embodiments of the present invention will bedescribed with reference to the accompanying drawings and tables.

[0075] First to Seventh Embodiments

[0076] First, as a first to a seventh embodiment of the invention,practical examples (Examples 1 to 7, respectively) of oblique projectionoptical systems will be presented below with reference to their actualconstruction data and other data. The oblique projection optical systems1 to 7 of Examples 1 to 7 are all composed of four poweredcurved-surface mirrors and one non-powered flat-surface mirror. Theseoblique projection optical systems 1 to 7 are all designed to lead raysof light from a rectangular display surface having longer sides in thewidth direction thereof to a projection surface by reflecting the rayswith the individual mirrors in such a way as to deflect the rays in theheight direction of the display surface and make the rays converge onthe projection surface. As a result, a magnified image of the imagedisplayed on the display surface is formed (projected) on the projectionsurface, in a rectangular area thereon that is substantially similar tothe display surface.

[0077] In each example, the display surface is represented by S0, andthe projection surface is represented by S6. The reflecting surfaces ofthe individual mirrors are represented by S1 to S5 in the order in whichthey receive light from the display surface S0 (i.e. in order ofproximity to the display surface S0 along the optical path). The pupilplane (aperture stop) is represented by APR.

[0078] The oblique projection optical systems 1 to 7 of Examples 1 to 7include, as all or part of the powered reflecting surfaces S1 to S4,free-form curved surfaces. Thus, any of these optical systems issymmetric about a plane, but has no optical axis that holds throughoutthe optical system. Accordingly, it is not proper to define the surfacesS0 to S6 in a coordinate system that uses an optical axis as areference. Instead, in an absolute coordinate system, local coordinatesystems are defined one for each of the surface S0 to S6, so that thesurfaces S0 to S6 are represented by their respective coordinatesystems. Then, the optical system as a whole is defined in terms of thepositions and directions of the individual local coordinate systems inthe absolute coordinate system.

[0079] Here, Cartesian coordinate systems are used as the absolute andlocal coordinate systems. The coordinate axes of the absolute coordinatesystems are referred to as the x-axis, y-axis, and z-axis, and thecoordinate axes of a local coordinate system are referred to as theX-axis, Y-axis, and Z-axis. All lengths are given in mm. The absolutecoordinate system has its origin at the center of the display surfaceS0, and has its x-, y-, and z-axes respectively in the direction normalto the projection surface S6, in the height direction thereof, and inthe width direction thereof. Each local coordinate system has its originon the x-y plane of the absolute coordinate system. For each localcoordinate system, the unit vectors along its X-, Y-, and Z-axes arerepresented respectively by VX, VY, and VZ, and these unit vectors VX,VY, and VZ are defined in the absolute coordinate system to define thedirection of the local coordinate system in the absolute coordinatesystem. The Z-axis of each local coordinate system is parallel to thez-axis of the absolute coordinate system, and therefore the X-Y planecoincides with the x-y plane. The surfaces S0 to S6 are symmetric aboutthe X-Y plane of their respective local coordinate systems, and theoptical system as a whole is symmetric about the x-y plane of theabsolute coordinate system.

[0080] The surfaces S0 to S6 are each defined by the formula of anextended aspherical surface ES below. $\begin{matrix}{X = {\frac{C\quad {0 \cdot H^{2}}}{1 + \left( {1 - {{ɛ \cdot C}\quad {0^{2} \cdot H^{2}}}} \right)^{1/2}} + {\sum\limits_{i}{{Ai} \cdot H^{i}}} + {\sum\limits_{j,k}{{Bjk} \cdot Y^{j} \cdot Z^{k}}}}} & ({ES})\end{matrix}$

[0081] In this formula, C0 represents the curvature at the vertex (theintersection with the X-axis); E represents the conic constant; i, j,and k each represent an integer number equal to or greater than 0; andH²=Y²+Z². Ai represents the coefficient for the term that includes H tothe power of i, and Bjk represents the coefficient for the term thatincludes Y to the power of j and Z to the power of k. In each example,terms including H to the power i of up to 12 are considered, and termsincluding Y and Z to the power j+k of 10 are considered. In thepresentation of each example, the values of the coefficients Ai and Bjkare given, but those of which the value equals 0 are omitted unlessnecessary.

EXAMPLE 1

[0082] FIGS. 1 to 4 show the construction and optical path of theoblique projection optical system 1 of Example 1, and Tables 3 to 10show the construction data thereof. Tables 3 to 10 each list the data ofthe surface referred to by the symbol noted at the top of the table. N0and N1 respectively represent the refractive indices of the media beforeincidence and after incidence (i.e. after reflection) on a surface.“Position” indicates the position of the origin of the correspondinglocal coordinate system in the absolute coordinate system. In Table 5,which lists the data of the pupil plane APR, R represents the radius ofthe pupil (aperture stop).

[0083] It is to be noted that, also in the other examples describedlater, what their construction data represents is the same as withTables 3 to 10.

[0084]FIG. 1 is a sectional view taken along the x-y plane, and showsthe surfaces S0 to S6 together with, among the rays emanating from thecenter in the width direction of the display surface S0, five rays, i.e.two emanating from both ends (end points) in the height direction of thedisplay surface S0 and three emanating from the three points that dividethe line between those ends into four equal parts. FIG. 2 is a side viewas seen from the z direction, and shows, in a form superposed on thefive rays mentioned above, among the rays emanating from both ends inthe width direction of the display surface S0, those emanating from thesame points in the height direction of the display surface S0 asdescribed above. Here, the surfaces are not marked with their symbols S0to S6.

[0085]FIG. 3 is a top view as seen from the y direction, and shows thesurfaces S0 to S6 together with, among the rays emanating from thecenter in the height direction of the display surface S0, nine rays,i.e. two emanating from both ends in the width direction of the displaysurface S0 and seven emanating from the seven points that divide theline between those ends into eight equal parts. FIG. 4 is a front viewas seen from the x direction, and shows the surfaces S0 to S6 togetherwith, among the rays emanating from both ends (end lines) in the heightdirection of the display surface S0 and from three lines that divide thearea between those ends into four equal parts, 45 rays in total, i.e.,for each of these five lines, two emanating from both ends in the widthdirection of the display surface S0 and seven emanating from the sevenpoints that divide the line between those ends into eight equal parts.

[0086]FIG. 5 shows a spot diagram obtained on the projection surface S6,more specifically, near the intersections between, on the one hand, bothends (end lines) in the height direction of the projection surface S6and the lines that divide the area between those ends into four equalparts and, on the other hand, both ends (end lines) in the widthdirection of the projection surface S6 and the lines that divide thearea between those ends into eight equal parts. These intersectionscorrespond to the origins of the coordinate axes shown in the individualsections of the diagram. Since the optical system is symmetric in thewidth direction (the z direction), the obtained results are shown foronly half of the projection surface S6. That is, FIG. 5 is a diagram for25 among the 45 rays shown in FIG. 4, and the third-from-the-above,leftmost section of the diagram shows the results obtained near thecenter of the projection surface S6. In each section of the diagram,spots represent the positions in which different rays belonging to anidentical beam of light are incident. Above each section of the diagramare noted the coordinates (in the local coordinate system) of the centerof the incident positions of all the rays belonging to an identicalbeam. The values ±2 noted by the coordinate axes in each section of thediagram represent the distances from the origin of those coordinateaxes.

[0087]FIG. 6 shows the distortion of the image observed on theprojection surface S6. In this figure, solid lines represent the resultsobtained with the oblique projection optical system 1, and broken linesrepresent the ideal result without distortion.

[0088] It is to be noted that, also in the modified examples and theother examples described later, what their sectional view, top view,side view, spot diagram, and distortion diagram represent is the same aswith FIGS. 1 to 3, 5, and 6.

[0089] As shown in Tables 4 to 8, in the oblique projection opticalsystem 1 of this example, the reflecting surface S1 is a sphericalsurface, the reflecting surface S2 is an aspherical surface, thereflecting surface S3 is a free-form curved surface, and the reflectingsurface S4 is a free-form curved surface. The pupil plane APR is locatedbetween the reflecting surface S1 and the reflecting surface S2. Asshown in Tables 3, 9, and 10, the display surface S0, the reflectingsurface S5, and the projection surface S6 are flat surfaces, which areparallel to one another.

[0090] Table 1 shows the magnifications factors (the projectionmagnification factors) by which an image is magnified when projected,the sizes of the display surface S0 and the projection surface S6 (i.e.the sizes of the areas in which an image is displayed or projected), theangles of incidence at which rays are incident on the projection surfaceS6, and the position of the entrance pupil of the beam from the displaysurface S0 as observed in this example, together with the same data asobserved in the other examples. Here, the sizes of the display surfaceS0 and the projection surface S6 are given in half values. The positionof the entrance pupil is given as values of X and Y in the localcoordinate system of the display surface S0, with Y given as a lengthwhen X has a finite value, and with Y given as an angle when X isinfinite.

[0091] The magnification factors are those obtained from the center ofthe display surface S0 to the center of the projection surface S6, withβ(Y) representing the magnification factor in the height direction (theY, and thus y, direction), and with β(Z) representing the magnificationfactor in the width direction (the Z, and thus z, direction). Themagnification factor 0 calculated as the ratio of the size of theprojection surface S6 to the size of the display surface S0 isapproximately equal to those listed in the table. The magnificationfactors β(Y) and β(Z) in the height and width directions are almostequal to each other, and the slight difference between them is givenunder “Anamo(Y/Z).” It is to be noted that the reason that themagnification factor β(Z) in the width direction takes a negative valueis that the Z-axis of the local coordinate system points in the oppositedirection from one of the reflecting surfaces S1 to S5 to the next.

[0092] In the oblique projection optical system 1, the magnificationfactors β(Y) and β(Z) in the height and width directions are 78.57 and78.56 respectively. Rays are incident on the projection surface S6 atthe minimum angle of incidence (24.0°) at the lower end in the heightdirection at the center in the width direction, at the maximum angle ofincidence (67.3°) at the upper end in the height direction at both endsin the width direction, and at an angle of incidence of 52.1° at thecenter. Thus, the angle of view, which is defined as the differencebetween the maximum and minimum angles of incidence, is 43.3°. Theentrance pupil is located at infinity, making the optical system anoblique telecentric optical system.

[0093] Table 2 shows the f-numbers, the thickness D of the optical path,the length H in the height direction of the projection surface S6, theratio of the thickness D of the optical path to the length H of theprojection surface S6, the shapes of the powered reflecting surfaces S1to S4, and the signs of their powers as observed in this example,together with the same data as observed in the other example. Here, thef-numbers are effective f-numbers calculated from the diameter andposition of the entrance pupil. Fnoy represents the f-number in theheight direction of the display surface S0, and Fnoz represents thef-number in the width direction thereof. The thickness D of the opticalpath is the maximum length, in the direction normal to the projectionsurface S6 (i.e. in the X, and thus x, direction), of the space throughwhich light travels from the display surface S0 to projection surfaceS6.

[0094] The symbols “sp,” “asp,” and “exasp” respectively denotespherical, aspherical, and free-form curved surfaces. The symbol (+)denotes a concave surface having a positive power, and the symbol (−)denotes a convex surface having a negative power. Here, the powers ofthe reflecting surfaces S1 to S4 depend on the surface shape thereof atthe point at which the ray that travels from the center of the displaysurface S0 through the center of the pupil plane APR to the projectionsurface S6 passes therethrough, and not on the sign of the curvature C0in formula (ES) noted earlier by which the curved surfaces are defined.

[0095] In the oblique projection optical system 1, the f-numbers Fnoyand Fnoz in the height and width directions are 3.5 and 3.4respectively, and the ratio D/H of the thickness of the optical path tothe length in the height direction of the projection surface S6 is 0.30.The four reflecting surfaces S1 to S4 have, from the display surface S0side, a positive, a negative, a positive, and a negative power. Thus,the reflecting surface S4 closest to the projection surface S6 has anegative power.

[0096] Whereas the length H in the height direction of the projectionsurface S6 is about 685 mm, the length in the height direction of thepart of the optical system that is located below the lower end of theprojection surface S6 is about 210 mm, which is about 23% of the lengthin the height direction of the optical system as a whole. The distancebetween the projection surface S6 and the flat-surface reflectingsurface S5 is 200 mm, which is almost equal to the thickness D of theoptical path. The ratio of the length of the projection surface S6 inthe height direction to that in the width direction is about 9:16.

MODIFIED EXAMPLE 1 OF EXAMPLE 1

[0097]FIGS. 7 and 8 show a sectional view and a top view, respectively,of the oblique projection optical system 1-b of a modified example ofthe oblique projection optical system 1 of Example 1. The obliqueprojection optical system 1-b differs from its base model in that thesurfaces starting with the display surface S0 and ending with theflat-surface reflecting surface S5 are shifted toward the projectionsurface S6.

[0098] As will be clear from FIG. 7, the greater part of the reflectingsurface S4 is located opposite to the flat-surface reflecting surface S5with respect to the projection surface S6, and thus a lower centralportion of the oblique projection optical system 1-b protrudes a littlefrom the projection surface S6. As a result, as shown in Table 2described earlier, the thickness D of the optical path in the opticalsystem as a whole is larger, but, as will be understood throughcomparison between FIGS. 7 and 1, the distance between the projectionsurface S6 and the flat-surface reflecting surface S5 is shorter than inthe oblique projection optical system 1.

MODIFIED EXAMPLE 2 OF EXAMPLE 1

[0099]FIG. 9 shows a sectional view of the oblique projection opticalsystem 1-c of another modified example of the oblique projection opticalsystem 1 of Example 1. This oblique projection optical system 1-cdiffers from its base model in that the flat-surface reflecting surfaceS5 is omitted. As a result of the omission of the reflecting surface S5,as shown in Table 2, the thickness D of the optical path in the opticalsystem as a whole doubles. Still, the thickness D of the optical path isabout 59% of the length H in the height direction of the projectionsurface S6.

[0100] Whereas the oblique projection optical system 1 provided with thereflecting surface S5 is suitable only for rear projection, the obliqueprojection optical system 1-c is suitable for both rear projection andfront projection. It is to be understood that, in the oblique projectionoptical systems of the examples described below, the reflecting surfaceS5 may be omitted as in this modified example.

EXAMPLE 2

[0101] FIGS. 10 to 12 show a sectional view, a side view, and a topview, respectively, of the oblique projection optical system 2 ofExample 2, and Tables 11 to 18 show the construction data thereof FIG.13 shows a spot diagram obtained on the projection surface S6, and FIG.14 shows the distortion of the image observed on the projection surfaceS6.

[0102] As shown in Table 1, the magnification factors β(Y) and β(Z) inthe height and width directions are 71.41 and 71.39 respectively. Raysare incident on the projection surface S6 at the minimum angle ofincidence (24.7°) at the lower end in the height direction at the centerin the width direction, at the maximum angle of incidence (65.8°) at theupper end in the height direction at both ends in the width direction,and at an angle of incidence of 50.9° at the center. Thus, the angle ofview is 41.1°. The entrance pupil is located at infinity on the normalto the center of the display surface S0, making the optical system atelecentric optical system.

[0103] As shown in Table 2, the f-numbers Fnoy and Fnoz in the heightand width directions are 2.5 and 2.5 respectively, and the ratio D/H ofthe thickness of the optical path to the length in the height directionof the projection surface S6 is 0.32. The four reflecting surfaces S1 toS4 have, from the display surface S0 side, a positive, a negative, apositive, and a negative power. Thus, the reflecting surface S4 closestto the projection surface S6 has a negative power. The two reflectingsurfaces S1 and S2 close to the display surface S0 are asphericalsurfaces, and the two reflecting surfaces S3 and S4 close to theprojection surface S6 are free-form curved surfaces.

[0104] Whereas the length H in the height direction of the projectionsurface S6 is about 623 mm, the length in the height direction of thepart of the optical system that is located below the lower end of theprojection surface S6 is about 233 mm, which is about 27% of the lengthin the height direction of the optical system as a whole. The distancebetween the projection surface S6 and the flat-surface reflectingsurface S5 is 200 mm, which is almost equal to the thickness D of theoptical path. The ratio of the length of the projection surface S6 inthe height direction to that in the width direction is about 9:16.

EXAMPLE 3

[0105] FIGS. 15 to 17 show a sectional view, a side view, and a topview, respectively, of the oblique projection optical system 3 ofExample 3, and Tables 19 to 26 show the construction data thereof FIG.18 shows a spot diagram obtained on the projection surface S6, and FIG.19 shows the distortion of the image observed on the projection surfaceS6.

[0106] As shown in Table 1, the magnification factors β(Y) and β(Z) inthe height and width directions are 71.40 and 71.39 respectively. Raysare incident on the projection surface S6 at the minimum angle ofincidence (24.3°) at the lower end in the height direction at the centerin the width direction, at the maximum angle of incidence (65.7°) at theupper end in the height direction at both ends in the width direction,and at an angle of incidence of 50.7° at the center. Thus, the angle ofview is 41.4°. The entrance pupil is located at infinity, making theoptical system an oblique telecentric optical system.

[0107] As shown in Table 2, the f-numbers Fnoy and Fnoz in the heightand width directions are 2.6 and 2.5 respectively, and the ratio D/H ofthe thickness of the optical path to the length in the height directionof the projection surface S6 is 0.32. The four reflecting surfaces S1 toS4 have, from the display surface S0 side, a positive, a negative, apositive, and a negative power. Thus, the reflecting surface S4 closestto the projection surface S6 has a negative power. The two reflectingsurfaces S1 and S2 close to the display surface S0 are asphericalsurfaces, and the two reflecting surfaces S3 and S4 close to theprojection surface S6 are free-form curved surfaces.

[0108] Whereas the length H in the height direction of the projectionsurface S6 is about 623 mm, the length in the height direction of thepart of the optical system that is located below the lower end of theprojection surface S6 is about 225 mm, which is about 27% of the lengthin the height direction of the optical system as a whole. The distancebetween the projection surface S6 and the flat-surface reflectingsurface S5 is 200 mm, which is almost equal to the thickness D of theoptical path. The ratio of the length of the projection surface S6 inthe height direction to that in the width direction is about 9:16.

EXAMPLE 4

[0109] FIGS. 20 to 22 show a sectional view, a side view, and a topview, respectively, of the oblique projection optical system 4 ofExample 4, and Tables 27 to 34 show the construction data thereof. FIG.23 shows a spot diagram obtained on the projection surface S6, and FIG.24 shows the distortion of the image observed on the projection surfaceS6.

[0110] The pupil plane APR is located between the reflecting surface S1closest to the display surface S0 and the display surface S0, making theoblique projection optical system 4 a rear-aperture-type optical system.The aperture stop may be located between the reflecting surfaces S1 andS2 so as not to obstruct the beam traveling therebetween.

[0111] As shown in Table 1, the magnification factors β(Y) and β(Z) inthe height and width directions are 51.23 and 51.27 respectively. Raysare incident on the projection surface S6 at the minimum angle ofincidence (35.9°) at the lower end in the height direction at the centerin the width direction, at the maximum angle of incidence (71.4°) at theupper end in the height direction at both ends in the width direction,and at an angle of incidence of 61.3° at the center. Thus, the angle ofview is 35.5°. The entrance pupil is located at a finite distance fromthe display surface S0, making the optical system a non-telecentricoptical system.

[0112] As shown in Table 2, the f-numbers Fnoy and Fnoz in the heightand width directions are 3.6 and 3.5 respectively, and the ratio D/H ofthe thickness of the optical path to the length in the height directionof the projection surface S6 is 0.26. The four reflecting surfaces S1 toS4 have, from the display surface S0 side, a positive, a negative, apositive, and a negative power. Thus, the reflecting surface S4 closestto the projection surface S6 has a negative power. All these fourreflecting surfaces S1 to S4 are free-form curved surfaces.

[0113] Whereas the length H in the height direction of the projectionsurface S6 is about 498 mm, the length in the height direction of thepart of the optical system that is located below the lower end of theprojection surface S6 is about 228 mm, which is about 31% of the lengthin the height direction of the optical system as a whole. The distancebetween the projection surface S6 and the flat-surface reflectingsurface S5 is 110 mm, which is about 31 mm less than the thickness D ofthe optical path. Thus, a lower central portion of the optical systemprotrudes a little from the projection surface S6. The ratio of thelength of the projection surface S6 in the height direction to that inthe width direction is about 9:16.

[0114] In the oblique projection optical systems 1 to 3 of Examples 1 to3 described earlier, the beam traveling from the display surface S0 tothe reflecting surface S1 has a high degree of symmetry in the heightdirection of the display surface S0. Therefore, it is difficult toilluminate the panel that displays the image on the display surface S0from the reflecting surface S1 side. Thus, the oblique projectionoptical systems 1 to 3 are suitable for use with a transmissive imagedisplay panel, such as a transmissive liquid crystal panel, is used.

[0115] By contrast, in the oblique projection optical system 4 of thisexample, the beam traveling from the display surface S0 to thereflecting surface S1 shows striking asymmetry in the height directionof the display surface S0. Therefore, it is possible to illuminate theimage display panel from the reflecting surface S1 side. Thus, theoblique projection optical system 4 is suitable for use with both atransmissive panel and a reflective panel. As a reflective panel, it ispossible to use a reflective liquid crystal panel, or a mirror devicecomposed of a large number of minute mirror elements which modulatesillumination light by varying the direction of the individual mirrorelements.

EXAMPLE 5

[0116] FIGS. 25 to 27 show a sectional view, a side view, and a topview, respectively, of the oblique projection optical system 5 ofExample 5, and Tables 35 to 42 show the construction data thereof. FIG.28 shows a spot diagram obtained on the projection surface S6, and FIG.29 shows the distortion of the image observed on the projection surfaceS6.

[0117] As shown in Table 1, the magnification factors β(Y) and β(Z) inthe height and width directions are 51.25 and 51.27 respectively. Raysare incident on the projection surface S6 at the minimum angle ofincidence (34.8°) at the lower end in the height direction at the centerin the width direction, at the maximum angle of incidence (69.7°) at theupper end in the height direction at both ends in the width direction,and at an angle of incidence of 59.8° at the center. Thus, the angle ofview is 34.9°. The entrance pupil is located at a finite distance fromthe display surface S0, making the optical system a non-telecentricoptical system.

[0118] As shown in Table 2, the f-numbers Fnoy and Fnoz in the heightand width directions are 3.7 and 3.5 respectively, and the ratio D/H ofthe thickness of the optical path to the length in the height directionof the projection surface S6 is 0.24. The four reflecting surfaces S1 toS4 have, from the display surface S0 side, a positive, a negative, apositive, and a negative power. Thus, the reflecting surface S4 closestto the projection surface S6 has a negative power. All these fourreflecting surfaces S1 to S4 are free-form curved surfaces.

[0119] Whereas the length H in the height direction of the projectionsurface S6 is about 498 mm, the length in the height direction of thepart of the optical system that is located below the lower end of theprojection surface S6 is about 234 mm, which is about 32% of the lengthin the height direction of the optical system as a whole. The distancebetween the projection surface S6 and the flat-surface reflectingsurface S5 is 120 mm, which is equal to the thickness D of the opticalpath. The ratio of the length of the projection surface S6 in the heightdirection to that in the width direction is about 9:16.

EXAMPLE 6

[0120] FIGS. 30 to 32 show a sectional view, a side view, and a topview, respectively, of the oblique projection optical system 6 ofExample 6, and Tables 43 to 50 show the construction data thereof FIG.33 shows a spot diagram obtained on the projection surface S6, and FIG.34 shows the distortion of the image observed on the projection surfaceS6.

[0121] As shown in Table 1, the magnification factors β(Y) and β(Z) inthe height and width directions are 70.45 and 70.50 respectively. Raysare incident on the projection surface S6 at the minimum angle ofincidence (40.2°) at the lower end in the height direction at the centerin the width direction, at the maximum angle of incidence (73.2°) at theupper end in the height direction at both ends in the width direction,and at an angle of incidence of 64.3° at the center. Thus, the angle ofview is 33.0°. The entrance pupil is located very far away from thedisplay surface S0, making the optical system a non-telecentric opticalsystem close to a telecentric optical system.

[0122] As shown in Table 2, the f-numbers Fnoy and Fnoz in the heightand width directions are 3.4 and 3.3 respectively, and the ratio D/H ofthe thickness of the optical path to the length in the height directionof the projection surface S6 is 0.21. The four reflecting surfaces S1 toS4 have, from the display surface S0 side, a positive, a negative, apositive, and a negative power. Thus, the reflecting surface S4 closestto the projection surface S6 has a negative power. All these fourreflecting surfaces S1 to S4 are free-form curved surfaces.

[0123] Whereas the length H in the height direction of the projectionsurface S6 is about 685 mm, the length in the height direction of thepart of the optical system that is located below the lower end of theprojection surface S6 is about 289 mm, which is about 30% of the lengthin the height direction of the optical system as a whole. The distancebetween the projection surface S6 and the flat-surface reflectingsurface S5 is 145 mm, which is equal to the thickness D of the opticalpath. The ratio of the length of the projection surface S6 in the heightdirection to that in the width direction is about 9:16.

EXAMPLE 7

[0124] FIGS. 35 to 37 show a sectional view, a side view, and a topview, respectively, of the oblique projection optical system 7 ofExample 7, and Tables 51 to 58 show the construction data thereof FIG.38 shows a spot diagram obtained on the projection surface S6, and FIG.39 shows the distortion of the image observed on the projection surfaceS6.

[0125] As shown in Table 1, the magnification factors β(Y) and β(Z) inthe height and width directions are 51.24 and 51.27 respectively. Raysare incident on the projection surface S6 at the minimum angle ofincidence (36.8°) at the lower end in the height direction at the centerin the width direction, at the maximum angle of incidence (69.1°) at theupper end in the height direction at both ends in the width direction,and at an angle of incidence of 58.6° at the center. Thus, the angle ofview is 32.2°. The entrance pupil is located at infinity, making theoptical system a telecentric optical system.

[0126] As shown in Table 2, the f-numbers Fnoy and Fnoz in the heightand width directions are both 2.5, and the ratio D/H of the thickness ofthe optical path to the length in the height direction of the projectionsurface S6 is 0.31. The four reflecting surfaces S1 to S4 have, from thedisplay surface S0 side, a positive, a negative, a positive, and anegative power. Thus, the reflecting surface S4 closest to theprojection surface S6 has a negative power. All these four reflectingsurfaces S1 to S4 are free-form curved surfaces.

[0127] Whereas the length H in the height direction of the projectionsurface S6 is about 498 mm, the length in the height direction of thepart of the optical system that is located below the lower end of theprojection surface S6 is about 233 mm, which is about 32% of the lengthin the height direction of the optical system as a whole. The distancebetween the projection surface S6 and the flat-surface reflectingsurface S5 is 125 mm, which is about 27 mm less than the thickness D ofthe optical path. Thus, a lower central portion of the optical systemprotrudes a little from the projection surface S6. The ratio of thelength of the projection surface S6 in the height direction to that inthe width direction is about 9:16.

[0128] In Examples 1 to 7 described thus far, the display surface S0 andthe projection surface S6 are arranged parallel. However, the displaysurface S0 may be arranged so as to be inclined relative to theprojection surface S6. Such arrangement is easy in the obliqueprojection optical systems 1 to 7 provided with reflecting surfaceshaving free-form curved surfaces. In these examples, all the poweredsurfaces are reflecting surfaces. However, part of the powered surfacesmay be realized with refractive surfaces. That is, in the obliqueprojection optical systems 1 to 7, it is possible to use lenses incombination with mirrors, or to use lenses instead of the mirrors havingcurved or aspherical surfaces.

[0129] More than-one display surface may be provided; that is, it ispossible, by the use of a cross prism or the like, to provide aplurality of display surfaces that are optically equivalent to oneanother. For example, by arranging a cross dichroic prism between thedisplay surface S0 and the reflecting surface S1, it is possible toarrange two display surfaces equivalent to the display surface S0. Then,by displaying red, green, and blue components of an image on these threedisplay surfaces, and then integrating together the light of these colorcomponents with the cross dichroic prism, it is possible to form a colorimage on the projection surface S6. In any of the oblique projectionoptical systems 1 to 7, there is sufficient room to arrange such a crossprism in a portion of the space between the reflecting surface S1 andthe display surface S0 close to the display surface S0. It is to benoted that, even with a single display surface S0, it is possible topresent a color image by displaying red, green, and blue components ofan image thereon on a time division basis.

[0130] As described above, in the first to seventh embodiments, in anoblique projection optical system that projects a magnified image of animage displayed on a display surface onto a projection surface and thatis provided with a plurality of reflecting surfaces having a power, therays of light from the display surface on which the image is displayedare led to the projection surface in such a way that the ray from thecenter of the display surface is obliquely incident on the projectionsurface. At least two of the reflecting surfaces having a power arefree-form curved surfaces. Moreover, of the reflecting surfaces having apower, the one closest to the projection surface has a negative power,and at least one of the other reflecting surfaces has a positive power.

[0131] This oblique projection optical system, as opposed toconventional projection optical systems built as centered opticalsystems, adopts free-form curved surfaces as reflecting surfaces.Adopting free-form curved surfaces as reflecting surfaces makes itpossible to achieve oblique projection almost free from distortion witha short optical path length without sacrificing imaging performance. Byarranging a means for displaying an image on the display surface and ascreen on the projection surface, it is possible to obtain a projectionapparatus. Having a short optical path length, the projection apparatusthus obtained is not only slim, i.e. has a small dimension in thedirection perpendicular to the screen, but has a small dimension also inthe direction along the screen. When provided with a flat-surface mirrorfor turning the optical path, this oblique projection optical system issuitable for use in rear projection apparatus, but it can be used infront projection apparatus as well.

[0132] To obtain a sufficiently high magnification factor (projectionmagnification factor), it is preferable that, of the reflecting surfaceshaving a power, the one closest to the projection surface have anegative power. Accordingly, to permit the rays from different points onthe display surface to converge on one point on the projecting surface,at least one of the other reflecting surfaces having a power needs tohave a positive power. Thus, the powers of the reflecting surfaces aredetermined so as to fulfill these requirements.

[0133] The oblique projection optical system described above has fourreflecting surfaces having a power, and it is preferable that thesereflecting surfaces have a positive, a negative, a positive, and anegative power in order of proximity to the display surface. This makesit easy to shorten the optical path from the display surface to theprojection surface, and thus to slim down a projection apparatusincorporating the oblique projection optical system while simultaneouslyreducing the dimension of the projection apparatus in the directionalong the screen.

[0134] The oblique projection optical system may be so configured thatthe display surface has a smaller dimension in the height direction thanin the width direction, that the reflecting surfaces having a power eachreflect the rays of light from the display surface in such a way as todeflect the rays in the height direction of the display surface, thatthe pupil plane is located between the one of the reflecting surfaceshaving a power that is second-closest to the display surface and thedisplay surface, and that the following conditions are fulfilled:Fnoy≧Fnoz, Fnoy≦4.5, and Fnoz≦4.0, where Fnoy represents the f-number inthe direction corresponding to the height direction of the displaysurface, and Fnoz represents the f-number in the direction correspondingto the width direction of the display surface. Making each reflectingsurface reflect the rays from the display surface in such a way as todeflect the rays in the height direction of the display surface makes iteasy to reduce the size of a projection apparatus incorporating theoblique projection optical system in the height direction of the screen.Moreover, setting the position of the pupil plane and the relationshipbetween the f-numbers in this way make it possible to present brightimages.

[0135] Alternatively, the oblique projection optical system may be soconfigured that the display surface has a smaller dimension in theheight direction than in the width direction, the reflecting surfaceshaving a power each reflect the rays of light from the display surfacein such a way as to deflect the rays of light in the height direction ofthe display surface, and the following condition is fulfilled: D/H≦0.35,where H represents the dimension of the projection surface in thedirection corresponding to the height direction of the display surface,and D represents the maximum length, in the direction normal to theprojection surface, of the space through which the rays of light pass totravel from the display surface to the projection surface. A projectionapparatus incorporating this oblique projection optical system is slimrelative to the size of the screen.

[0136] Here, the following condition may additionally be fulfilled:30≦β≦100, where β represents the ratio of the size of the projectionsurface to the size of the display surface. The symbol β represents themagnification factor by which the image is magnified by projection. Witha magnification factor lower than 30, interference between thereflecting surfaces themselves needs to be avoided by shifting, in thedirection perpendicular to the projection surface, reflecting surfacesthat are adjacent to each other in space. On the other hand, with amagnification factor over 100, it is necessary to reduce the f-numbersto secure sufficient brightness, and thus it is difficult to obtainhigher imaging performance while shortening the optical length. Byfulfilling the condition described above, it is possible both tomaintain high imaging performance and to slim down and miniaturize aprojection apparatus incorporating this oblique projection opticalsystem.

[0137] Eighth to Tenth Embodiments

[0138] Next, the rear projection optical systems of an eighth to a tenthembodiment of the invention will be described. FIGS. 40, 44, and 48 showthe entire projection path from a panel display surface I1 to a screensurface 12 in the eighth, ninth, and tenth embodiments, respectively.FIGS. 41, 45, and 49 show, in enlarged views, the projection opticalsystem constituting the eighth, ninth, and tenth embodiments,respectively, and a principal portion of the optical path thereof. Theseoptical path diagrams show optical sections taken along the Y-Z plane ofthe Cartesian coordinate system (X, Y, and Z) described later. It is tobe understood that the rear projection optical systems of theseembodiments do not necessarily have to be designed precisely as shown intheir respective optical path diagrams, but may be designed upside down;that is, turning their construction upside down to suit actualarrangement does not affect their function in any way.

[0139] The eighth to tenth embodiments deal with rear projection opticalsystems for use in rear-projection-type image projection apparatus (rearprojectors). These rear projection optical systems are provided with aprojection optical system for projecting a magnified image of atwo-dimensional image displayed on a panel display surface I1 (the imagedisplay surface of a display panel located on the reduction side) onto ascreen surface I2. The display panel is realized with a display devicesuch as a reflective liquid crystal panel, transmissive liquid crystalpanel, or DMD (digital micromirror device). The panel display surface I1is illuminated with illumination light emitted from a lamp (not shown)and passing through an illumination optical system (not shown). As thepanel display surface I1 is illuminated, projection light emanatestherefrom, which is then led to the screen surface I2 by the projectionoptical system and other components described later. Projection of acolor image is achieved by adopting a three-panel construction in whichthree display panels are arranged and color integration is achieved bythe use of a cross dichroic prism or the like, a single-panelconstruction in which an image is displayed on a time division basis, ora single-panel construction in which a microlens array is arranged on adisplay panel.

[0140] The rear projection optical systems of the eighth to tenthembodiments include, from the panel display surface I1 side, aprojection optical system composed of a first to a fourth mirror M1 toM4 and an optical path turning mirror composed of a fifth and a sixthmirror M5 and M6. In all these embodiments, the mirrors constituting theprojection optical system are all curved-surface reflecting mirrors, ofwhich reflecting surfaces are all free-form curved surfaces. Moreover,in all these embodiments, the two mirrors constituting the optical pathturning mirror are both flat-surface reflecting mirrors. The projectionlight emanating from the panel display surface I1 is reflected by thefour curved-surface reflecting mirrors constituting the projectionoptical system, then has its optical path turned twice by the twoflat-surface reflecting mirrors, and then reaches the screen surface I2.The symbol ST represents an aperture stop position ST, which correspondsto a virtual aperture stop plane.

[0141] In all these embodiments, projection of a color image isachieved, as described above, by arranging a color integrating prism,such as a cross dichroic prism, near the screen surface I2. For example,illumination light is separated into R, G, and B light by anillumination optical system so as to be separately shone on threedisplay panels and then integrated back together by a cross dichroicprism. The cross dichroic prism may be used for both color separationand color integration. When the display panel is of a reflective type,incident and reflected rays may be separated by the use of a beamseparating prism, such as a polarization beam splitter (PBS) or TIR(total internal reflection) prism. A condenser lens may be arranged nearthe panel display surface I1 to make the rear projection optical systemtelecentric toward the panel display surface I1.

[0142] In a case where a display panel, such as a liquid crystal panel,that exhibits different characteristics depending on the angle ofincidence thereon is used, it is preferable to make the projectionoptical system telecentric toward the panel display surface I1. However,to obtain higher optical performance, it is better to make it lesstelecentric. Thus, in a non-telecentric optical system, a condenser lensmay be arranged in front of the panel display surface I1 to make theoptical system telecentric with respect to the panel display surface I1.In a case where a reflective display panel is used and beam splittingneeds to be achieved without the use of a PBS, to permit incident andreflected rays to be separated on the basis of the difference betweentheir angles, the incident rays need to be inclined relative to thepanel display surface I1 at an angle larger than the angle determined bythe f-number in the direction in which the optical path is turned. Inthis case, it is preferable to adopt a so-called oblique telecentricconstruction (in which rays are incident obliquely on the panel displaysurface I1, at almost uniform angles of incidence over the entire areathereof). In an oblique telecentric construction, the aforementionedangle characteristics of liquid crystal do not matter. An obliquetelecentric construction may be realized by arranging a condenser lens(decentered as required) in front of the panel display surface I1.

[0143] An oblique projection optical system can be of one of thefollowing six types:

[0144] (i) a transmissive optical system employing part of a centeredoptical system;

[0145] (ii) a non-axis-symmetric transmissive optical system employing arelay;

[0146] (iii) a non-axis-symmetric transmissive optical system employingno relay;

[0147] (iv) a reflective optical system employing part of a centeredoptical system;

[0148] (v) a non-axis-symmetric reflective optical system; and

[0149] (vi) a non-axis-symmetric optical system partly reflective andpartly transmissive.

[0150] With the type (i), to obtain a large oblique projection angle asin the eighth to tenth embodiments, the original centered optical systemneeds to have a very wide angle of view. In general, attempting toobtain satisfactory optical performance with a wide-angle lens resultsin using many lenses and thus in high cost. With the type (ii), i.e. anon-axis-symmetric optical system that employs a relay to eliminatetrapezoid distortion, it is necessary to form an intermediary image.This makes the projection optical system very large. With the types(iii) and (vi), oblique projection is achieved by the use of, forexample, free-form curved surfaces or the like. However, sincetransmissive optical components cause dispersion and thus chromaticaberration, it is necessary to use additional optical components tocorrect it. Thus, even more components need to be used than with thereflective types (iv) and (v). With the type (iv), no chromaticaberration appears, but, as with the type (i), the centered opticalsystem requires a very large number of lenses.

[0151] With the type (v), the reflective optical system does not causechromatic aberration. Moreover, by using free-form curved surfaces thatare decentered relative to each other, it is possible to obtainsatisfactory optical performance and distortion-free images, whichcannot be achieved with a centered optical system. To achieve this, asin the eighth to tenth embodiments, it is preferable that the projectionoptical system have at least three curved-surface reflecting mirrors,and, for maximum compactness, it is further preferable that theprojection optical system form no intermediary image in the optical pathfrom the panel display surface I1 to the screen surface I2. In general,in a reflective optical system, it is necessary to use at least onepositive and one negative reflecting mirror to correct for the Petzvalsum, and it is necessary to use another free-form curved-surface mirrorto correct for distortion; that is, using at least three reflectingmirrors in total makes it possible to realize a projection opticalsystem that offers satisfactory optical performance and that producesalmost distortion-free images.

[0152] When the projection optical system is composed of threereflecting mirrors, i.e. the least required as described above, it maybe possible to obtain satisfactory optical performance, but theprojection optical system is then extremely sensitive to errorsinevitable in the assembly process when it is manufactured. That is, itsoptical performance deteriorates greatly by going through the assemblyprocess. To avoid this, it is preferable that, as in the eighth to tenthembodiments, the projection optical system have at least fourcurved-surface reflecting mirrors. Using at least four curved-surfacereflecting mirrors helps alleviate the responsibility of each reflectingsurface for the correction of aberrations, and helps disperse thesensitivity to assembly errors. Thus, as compared with a projectionoptical system composed of the least required number of reflectingmirrors, it is possible to reduce assembly errors

[0153] By giving at least three of the curved-surface reflecting mirrorsa free-form curved surface, it is possible to obtain better opticalperformance. Therefore, in a reflective optical system, like those ofthe eighth to tenth embodiments, that has four or more curved-surfacereflecting mirrors, it is preferable that at least three of thosecurved-surface reflecting mirrors have a free-form curved surface. Here,a free-form curved surface denotes a surface that includes a greatlydecentered aspherical surface and that does not have an axis of rotationsymmetry near the center of its effective area, that is, a surface thatis not spherical but has aspherical undulations (freedom). Theaspherical undulations of a free-form curved surface can be exploited tocontrol the curvature of a reflecting surface three-dimensionally. Thispermits non-axis-symmetric aberrations (distortions and the like)resulting from oblique projection to be corrected for easily withsurface inclinations that are so set as to vary from point to point onthe reflecting surface.

[0154] It is further preferable that the free-form curved surfaces usedas the reflecting surfaces of the curved-surface reflecting mirrors haveno axis of rotation symmetry but one plane of symmetry. In the eighth totenth embodiments, the Y-Z plane (the plane parallel to the plane oftheir respective optical path diagrams) is the plane of symmetry of eachfree-form curved surface. That is, the reflecting surface of eachcurved-surface reflecting mirror is a free-form curved surface that issymmetric about that plane of symmetry. Free-form curved surfaces likethese that are symmetric about a plane are easier to produce andevaluate than those which are not symmetric about a plane.

[0155] Moreover, in the eighth to tenth embodiments, the rear projectionoptical system as a whole is symmetric about the plane (i.e. the Y-Zplane) that runs vertically through the center of the screen. This makesthe production of optical components easy, and helps alleviate unevenbrightness and uneven distortion between the right and left parts of thescreen. However, where compactness matters as in a rear projectiontelevision set, the rear projection optical system as a whole can bemade smaller by turning the optical path in the width direction (i.e.the X direction). For example, by arranging, between the curved-surfacereflecting mirror that serves as the last surface of the projectionoptical system and the flat-surface reflecting mirror that is the firstto reflect the light exiting from the projection optical system, aflat-surface reflecting mirror that turns the optical path in the widthdirection (the X direction), it is possible to reduce the protrusion inan upper or lower portion of the rear projection television set (i.e.where the display panel and the projection optical system are arranged).Here, the optical path is turned with a flat-surface reflecting mirror,and therefore optical symmetry is not affected.

[0156] In the eighth to tenth embodiments, the optical path is turnedonly in the direction perpendicular to the direction of the longer sides(i.e. the X direction) of the screen surface I2, that is, in thedirection parallel to the Y-Z plane. In optical arrangements, like thoseof these embodiments, where the optical path is turned with a pluralityof reflecting mirrors, the optical path needs to be turned so that raysdo not overlap. However, from the viewpoint of optical performance, itis not preferable to secure a large margin for the turning of theoptical path, because this increases the degree of non-axis symmetry. Itis possible to obtain satisfactory optical performance by reducing thecross sectional area of the beam in the direction in which the opticalpath is turned. The cross sectional area of the beam in the direction inwhich the optical path is turned is reduced, preferably, by making theaperture stop ST elliptic. That is, the diameter of the aperture stop inthe direction in which the optical path is turned (i.e. in the directionparallel to the Y-Z plane) is reduced, and the diameter of the aperturestop in the direction perpendicular thereto (i.e. in the X direction) isincreased. Using an elliptic aperture stop like this makes it possibleto realize a rear projection optical system that offers satisfactoryoptical performance (i.e. in which the turning of the optical pathcauses little non-axis symmetry) without changing the total area (andthus the brightness) of the aperture stop.

[0157] Assume that the ray traveling from the center of the paneldisplay surface I1 through the center of the aperture stop ST to thecenter of the screen surface I2 is called the “screen center ray.” Then,it is preferable that conditional formulae (1) and (2) below befulfilled. In a rear projection optical system, like those of the eighthto tenth embodiments, that includes a projection optical system havingat least four curved-surface reflecting mirrors, designing the opticalsystem to fulfill conditional formulae (1) and (2) makes it possible torealize a rear projection optical system that offers satisfactoryoptical performance with little distortion but is neverthelessadvantageous in terms of mass production and cost and that is slim andis composed of compact optical components such as reflecting mirrors.

0.5<DL/HL<3.5  (1)

10°<θ<70°  (2)

[0158] where

[0159] DL represents the optical distance traveled by the screen centerray from the last surface of the projection optical system to the screensurface I2;

[0160] HL represents the dimension of the screen surface I2 in thedirection parallel to the plane (corresponding to the Y-Z plane in theoptical path diagrams) formed at the center of the screen surface I2 bya normal to the screen surface I2 and the screen center ray (that is,this dimension corresponds to the length of the shorter sides of thescreen surface I2 in the eighth to tenth embodiments); and

[0161] θ represents the angle of incidence at which the screen centerray is incident on the screen surface I2.

[0162] Conditional formula (1) defines the preferable angle of view asthe ratio of the object distance (i.e. the projection distance) DL tothe size of the screen surface I2. If the lower limit of conditionalformula (1) is transgressed, a wide angle of view is required, andtherefore it is difficult to obtain satisfactory optical performance. Toobtain satisfactory optical performance, it is necessary to make theprojection optical system as a whole longer and use larger reflectingmirrors, or to increase the number of reflective optical componentsused. However, either remedy leads to higher cost and is thusundesirable. Moreover, in optical arrangements, like those of the eighthto tenth embodiments, that employ two flat-surface reflecting mirrors,it is essential to secure a certain object distance DL to the screensurface I2 to permit the optical path to be turned in a compact form. Ifconditional formula (1) is fulfilled, the optical path can be turnedwith the flat-surface reflecting mirror arranged on the screen surfaceI2 side of the projection optical system. This helps make the opticalsystem as a whole, including the screen surface I2, slim and compactwithout unduly increasing cost. It is preferable to fulfill conditionalformula (1) with its lower limit raised to 2.5. This makes it possibleto realize a slim projection optical system that offers better opticalperformance and that employs inexpensive optical components.

[0163] If the upper limit of conditional formula (1) is transgressed,the angle of view is narrow. This is advantageous from the viewpoint ofoptical performance, but makes the object distance DL to the screensurface I2 unnecessarily long, making the rear projection optical systemas a whole large. It is preferable to fulfill conditional formula (1)with its upper limit lowered to 3.2. This makes it possible to realize amore compact rear projection optical system.

[0164] Conditional formula (2) defines the preferable oblique projectionangle. If the upper limit of conditional formula (2) is transgressed,the oblique projection angle is very large. A large oblique projectionangle makes it difficult to obtain satisfactory optical performance. Itis preferable to fulfill conditional formula (2) with its upper limitlowered to 63. This makes it possible to obtain better opticalperformance.

[0165] If the lower limit of conditional formula (2) is transgressed, itis easy to obtain satisfactory optical performance. However, rays arethen incident on the screen surface I2 from a direction close toperpendicular thereto. This makes it difficult to achieve slimming-downthrough oblique projection. It is preferable to fulfill conditionalformula (2) with its lower limit raised to 30. This makes it possible torealize a slimmer rear projection optical system. It is furtherpreferable to fulfill conditional formula (2) with its lower limitraised to 40. This makes it possible to achieve further slimming-down.It is even further preferable to fulfill conditional formula (2) withits lower limit raised to 45. This makes it possible to achieve evenfurther slimming-down.

[0166] Where, as in the eighth to tenth embodiments, an obliqueprojection optical system is used, it is preferable to use a screensuitable for the particular rear projection optical system used torealize it. Typically, with a rear projection television set or the likeis used a screen having a Fresnel lens, a lenticular plate, and a blackmatrix arranged in this order from the incident side. In obliqueprojection, a ray is incident at an angle on the center of the screensurface I2, and therefore, as shown in FIG. 52, it is preferable eitherto use a decentered Fresnel lens FL having a flat-surface portion FA onits side on which the projection light is incident (in the figure, theelements other than the Fresnel lens FL are omitted), or to use a screencomposed of a total reflection prism array and an ordinary Fresnel lenscombined together. If, contrary to the arrangement shown in FIG. 52, aFresnel portion (FB) is located on the side on which the projectionlight is incident, vignetting occurs.

[0167] With the arrangement shown in FIG. 52, rays appear discontinuousat intervals equal to the pitch of the Fresnel lens. To alleviate thiseffect, it is preferable to make the pitch of the Fresnel lenssufficiently finer than the size of the pixels displayed on the screensurface I2. Specifically, it is preferable that the following conditionbe fulfilled: [the pitch of the Fresnel lens]/[the size of the pixels onthe screen]<0.5. It is further preferable that the following conditionbe fulfilled: [the pitch of the Fresnel lens]/[the size of the pixels onthe screen]<0.3. In the eighth to tenth embodiments, the problemdescribed above can be overcome by using, for example, a screen with apixel size of about 1 mm and a Fresnel lens pitch of about 0.2 mm.

[0168] Next, the relationship between the material of the mirrors andthe temperature characteristics of the rear projection optical system.Usually, a projector has a heat generating member in the form of a lightsource, and the individual optical components not only transmit orreflect light but also absorb a slight amount of light. Therefore, afterthe lamp is turned on, the temperature of those optical componentsrises. Moreover, the ambient temperature is never constant. Thus, it isdesired that a rear projection optical system offer satisfactory opticalperformance stably against variations in temperature. Moreover, it isgenerally known that the sensitivity to errors of reflective opticalcomponents such as reflecting mirrors is more than twice as high as thatof ordinary transmissive optical components. Therefore, in the eighth totenth embodiments, it is preferable that the curved-surface reflectingmirrors have their substrate made of glass, which exhibits relativelysmall variations in properties against variations in temperature. It ispreferable that the substrate is coated with a reflective coating suchas an enhanced reflective coating formed by vapor-depositing aluminum orsilver, or a reflective coating formed of dielectric multilayer film.However, aluminum and silver absorb a slight amount of light and thuspose a risk of generating heat. Thus, from the viewpoint of minimizingheat generation, it is preferable to use a reflective coating formed ofdielectric multilayer film.

[0169] The curved-surface reflecting mirror closest to the screensurface I2 has a relatively weak optical power and thus has lowsensitivity to errors. Therefore, this curved-surface reflecting mirrormay have its substrate made of plastic, such as PMMA (polymethylmethacrylate, PC (polycarbonate), or polyolefin resin. That is, areflecting mirror of which the substrate is made of plastic and coatedwith an enhanced reflective coating formed by vapor-depositing aluminumor silver may be used as the curved-surface reflecting mirror closest tothe screen surface I2, because this has little effect on the opticalperformance obtained. Moreover, using a plastic substrate instead of aglass substrate helps reduce cost. Considering the relationshipdescribed above between the material of the mirrors and the temperaturecharacteristics of the rear projection optical system, it is preferablethat at least the first and second curved-surface reflecting mirrors ascounted from the panel display surface I1 side have their substrate madeof glass.

[0170] Next, how focusing and zooming are achieved in the rearprojection optical system will be described. It is preferable to achievefocusing by moving the display panel along the screen center ray, or bymoving the first or second mirror M1 or M2 along it. It is preferable toachieve zooming by moving at least two curved-surface reflectingmirrors. It is to be noted that, in a rear projection television set,the screen surface I2 is kept in a fixed position, and therefore, toadapt the display area to the screen surface I2, it is necessary toadjust the display area within a margin of a few percent by zooming.

EXAMPLES 8 TO 10

[0171] Practical examples (Examples 8 to 10) of the eighth to tenthembodiments will be presented in detail below with reference to theirconstruction data, spot diagrams, and other data. Examples 8 to 10presented below correspond to the eighth to tenth embodiments,respectively, and therefore the figures showing those embodiments alsoshow the construction and optical path of Examples 8 to 10.

[0172] Tables 59 to 64, 65 to 70, and 71 to 76 show the constructiondata of Examples 8 to 10, respectively. Of these tables, Tables 59, 65,and 71 show the size (mm) of the panel display surface I1, the size (mm)of the screen surface I2, and the f-numbers (FNO) in the directions ofthe longer and shorter sides of the screen (the X and Y directions,respectively). Tables 60, 66, and 72 show the data of the individualsurfaces of the respective rear projection optical systems, in orderfrom the reduction side. Tables 61, 67, and 73 show the free-form curvedsurface data representing the shape of the curved surface of the firstmirror (M1). Tables 62, 68, and 74 show the free-form curved surfacedata representing the shape of the curved surface of the second mirror(M2). Tables 63, 69, and 75 show the free-form curved surface datarepresenting the shape of the curved surface of the third mirror (M3).Tables 64, 70, and 76 show the free-form curved surface datarepresenting the shape of the curved surface of the fourth mirror (M4).The data of each surface is given in coordinates (X, Y, and Z) in aright-handed Cartesian coordinate system. Specifically, the position (X,Y, and Z coordinates) of a surfaces is given as the coordinates (mm) ofits vertex in the coordinate of which the origin (0, 0, 0) is located atthe center of the screen surface I2, and the inclination (X, Y, and Zrotation) of the surface is given as the rotation angles (°) about theX, Y, and Z axes with respect to its vertex. In the optical pathdiagrams, the X axis runs vertically to the plane of the diagrams (thedirection pointing from the front to back side of the plane of thediagrams as seen from the viewer is the positive direction, and thecounter-clockwise rotation on the plane of the diagrams as seen from theviewer is the positive X rotation). The Y axis runs along theintersection line between the screen surface I2 and the plane of thediagrams (the upward direction in the diagrams is the positivedirection), and the Z axis runs along the normal to the screen surfaceI2 (the rightward direction in the diagrams is the positive direction).

[0173] The reflecting surface of each curved-surface reflecting mirroris a free-form curved surface, of which the surface shape is defined byformula (FS) below using coordinates (x, y, and z) in the Cartesiancoordinate system having its origin at the vertex of the surface. Table77 lists the values of the conditional formulae and the related data asobserved in each example. $\begin{matrix}{z = {{\left( {c \cdot h^{2}} \right)/\left\lbrack {1 + \sqrt{1 - {\left( {1 + K} \right) \cdot c^{2} \cdot h^{2}}}} \right\rbrack} + {\sum\limits_{m}{\sum\limits_{n}\left\lbrack {{C\left( {m,n} \right)} \cdot x^{m} \cdot y^{n}} \right\rbrack}}}} & ({FS})\end{matrix}$

[0174] where

[0175] z represents the displacement from the reference surface alongthe optical axis at the height of h;

[0176] h represents the height in the direction perpendicular to theoptical axis (h²=x²+y²)

[0177] c represents the paraxial curvature (=the reciprocal of theradius of curvature);

[0178] K represents the conic constant, and

[0179] C(m, n) represents the free-form surface coefficients (thosewhich equal zero are omitted).

[0180] The optical performance of Examples 8 to 10 is shown in spotdiagrams in FIGS. 42, 46, and 50 and distortion diagrams in FIGS. 43,47, and 51, respectively. The spot diagrams show the imagingcharacteristics (mm) observed on the screen surface I2 for light havinga wavelength of 550 (nm), and the distortion diagrams show the positions(mm), observed on the screen surface I2, of the rays corresponding to arectangular grid pattern displayed on the panel display surface I1. Inthe distortion diagrams, D1 (solid lines) indicates the distortion gridof each example, and D0 (broken lines) indicates the grid of ideal imagepoints (without distortion) calculated in consideration of theanamorphic ratio. The object height (mm) corresponding to each fieldposition is given in coordinates (x, y) in the coordinate system ofwhich the origin is located at the center of the panel display surfaceI1, of which the x axis runs in the same direction as the X axis, and ofwhich the y axis runs perpendicularly to the x axis and parallel to thepanel display surface I1. On the other hand, the image height (mm)corresponding to each field position is given in coordinates (x′, y′) inthe coordinate system of which the origin is located at the center ofthe screen surface I2, of which the x′ axis runs in the same directionas the X axis, and of which the y′ axis runs perpendicularly to the x′axis and parallel to the screen surface I2. Thus, the distortiondiagrams show the distortion (though only in the x′ negative direction)of the image actually projected on the screen surface I2 as observedfrom the direction perpendicular to the x′-y′ plane. TABLE 1 Examples 1to 7 Overall (1) Angle of Incidence on Projection Surface Size (Half)Center Dia− Magnifi− Magnifi− Projection Surface Bottom gonal cation βcation β Anamo Display Surface S0 S6 (Mini− (Maxi− Angle Entrance PupilExample (Y) (Z) (Y/Z) Height Width Height Width mum) mum) Center of ViewX Y 1 78.57 −78.56 −0.02% 4.36 7.75 342.4 608.8 24.0 67.3 52.1 43.3 ∞ −9.9* 2 71.41 −71.39 −0.02% 4.36 7.75 311.3 553.5 24.7 65.8 50.9 41.1 ∞ 0.0* 3 71.40 −71.39 −0.02% 4.36 7.75 311.3 553.5 24.3 65.7 50.7 41.4 ∞−11.3* 4 51.23 −51.27 0.08% 4.86 8.63 249.1 442.8 35.9 71.4 61.3 35.5 40−8 5 51.25 −51.27 0.03% 4.86 8.63 249.1 442.8 34.8 69.7 59.8 34.9 80 −206 70.45 −70.50 0.08% 4.86 8.63 342.4 608.8 40.2 73.2 64.3 33.0 1400 −2507 51.24 −51.27 0.06% 4.86 8.63 249.1 442.8 36.8 69.1 58.6 32.2 ∞  0.0*

[0181] TABLE 2 Examples 1 to 7 Overall (2) Optical Path ProjectionThickness Surface Reflecting Reflecting Reflecting Reflecting ExampleFnoy Fnoz D Height H D/H Surface S1 Surface S2 Surface S3 Surface S4 13.5 3.4 202.4 684.9 0.30 sp(+) asp(−) exasp(+) exasp(−) 1−b 3.5 3.4227.4 684.9 0.33 sp(+) asp(−) exasp(+) exasp(−) 1−c 3.5 3.4 402.4 684.90.59 sp(+) asp(−) exasp(+) exasp(−) 2 2.5 2.5 200.0 622.6 0.32 asp(+)asp(−) exasp(+) exasp(−) 3 2.6 2.5 200.0 622.6 0.32 asp(+) asp(−)exasp(+) exasp(−) 4 3.6 3.5 131.3 498.1 0.26 exasp(+) exasp(−) exasp(+)exasp(−) 5 3.7 3.5 120.0 498.1 0.24 exasp(+) exasp(−) exasp(+) exasp(−)6 3.4 3.3 145.0 684.9 0.21 exasp(+) exasp(−) exasp(+) exasp(−) 7 2.5 2.5152.3 498.1 0.31 exasp(+) exasp(−) exasp(+) exasp(−)

[0182] TABLE 3 Example 1 Display Surface S0 N0 = N1 = 1 LocalCoordinates x y z Position 0.0000 0.0000 0.0000 Vector VX 1.0000 0.00000.0000 VY 0.0000 1.0000 0.0000 VZ 0.0000 0.0000 1.0000 C0 0.000000

[0183] TABLE 4 Example 1 Reflecting Surface S1 N0 = N1 = 1 LocalCoordinates x y z Position 85.2186 0.4793 0.0000 Vector VX 0.9965 0.08380.0000 VY −0.0838 0.9965 0.0000 VZ 0.0000 0.0000 1.0000 C0 −0.008558

[0184] TABLE 5 Example 1 Pupil Plane APR N0 = N1 = 1 Local Coordinates xy z Position 28.0000 −19.5000 0.0000 Vector VX −1.0000 0.0000 0.0000 VY0.0000 1.0000 0.0000 VZ 0.0000 0.0000 −1.0000 C0 0.000000 R 8.900000

[0185] TABLE 6 Example 1 Reflecting Surface S2 N0 = N1 = 1 LocalCoordinates x y z Position −30.7050 −9.7728 0.0000 Vector VX −0.99960.0271 0.0000 VY 0.0271 0.9996 0.0000 VZ 0.0000 0.0000 −1.0000 C00.007362 ε A4 A6 A8 A10 A12 1.0 6.16741 × 10⁻⁷ 5.21815 × 10⁻¹⁰ −1.02686× 10⁻¹² 2.00142 × 10⁻¹⁵ −1.17276 × 10⁻¹⁸

[0186] TABLE 7 Example 1 Reflecting Surface S3 N0 = N1 = 1 LocalCoordinates x y z Position 84.4174 3.7185 0.0000 Vector VX 0.9957 0.09260.0000 VY −0.0926 0.9957 0.0000 VZ 0.0000 0.0000 1.0000 C0 0.007444 ε A4A6 A8 A10 A12 1.0 7.06885 × 10⁻⁸ −2.37725 × 10⁻¹¹ −326077 × 10⁻¹⁵1.19913 × 10⁻¹⁹ 0.00000 × 10⁰ Bjk j = 0 j = 1 j = 2 j = 3 j = 4 j = 5 k= 0  0.00000 × 10⁰  0.00000 × 10⁰ −7.41333 × 10⁻³ −4.06175 × 10⁻⁵−3.54978 × 10⁻⁷  2.15348 × 10⁻⁹ k = 2 −5.07015 × 10⁻³ −8.65237 × 10⁻⁶−8.90477 × 10⁻⁷ −1.88595 × 10⁻⁸ −2.85271 × 10⁻¹⁰ −5.41109 × 10⁻¹² k = 4 1.57125 × 10⁻⁷ −2.08221 × 10⁻⁹ −6.76821 × 10⁻¹⁰ −2.41217 × 10⁻¹¹−3.12921 × 10⁻¹³ −2.11459 × 10⁻¹⁵ k = 6 −3.66527 × 10⁻¹⁰ −2.29003 ×10⁻¹¹ −4.84278 × 10⁻¹³ −4.74385 × 10⁻¹⁵ −1.79513 × 10⁻¹⁷ k = 8  3.87171× 10⁻¹⁵  8.96657 × 10⁻¹⁷ −2.99122 × 10⁻¹⁹ k = 10  4.72013 × 10⁻¹⁹ Bjk j= 6 j = 7 j = 8 j = 9 j = 10 k = 0  2.06937 × 10⁻¹¹ −1.11585 × 10⁻¹²−1.32674 × 10⁻¹⁴ −1.00000 × 10⁻¹⁶ −3.57818 × 10⁻¹⁹ k = 2 −4.87903 ×10⁻¹⁴ −4.14794 × 10⁻¹⁶ −1.81052 × 10⁻¹⁸ k = 4 −6.34071 × 10⁻¹⁸

[0187] TABLE 8 Example 1 Reflecting Surface S4 N0 = N1 = 1 LocalCoordinates x y z Position −74.6145 15.5740 0.0000 Vector VX −0.93220.3619 0.0000 VY 0.3619 0.9322 0.0000 VZ 0.0000 0.0000 −1.0000 C0−0.005995 ε A4 A6 A8 A10 A12 1.0 −1.69975 × 10⁻⁶ 2.14404 × 10⁻¹⁰−1.29314 × 10⁻¹⁵ −1.08992 × 10⁻¹⁹ 3.57834 × 10⁻²⁴ Bjk j = 0 j = 1 j = 2j = 3 j = 4 j = 5 k = 0  0.00000 × 10⁰  0.00000 × 10⁰ −4.35314 × 10⁻²−1.23693 × 10⁻³ −9.02155 × 10⁻⁶  4.57518 × 10⁻⁸ k = 2  6.98217 × 10⁻³−4.70094 × 10⁻⁴ −7.53704 × 10⁻⁶ −4.73627 × 10⁻⁸  9.21703 × 10⁻¹³ 1.20840 × 10⁻¹² k = 4 −2.22242 × 10⁻⁶ −3.08825 × 10⁻⁸  5.63301 × 10⁻¹⁰ 1.42978 × 10⁻¹¹ −8.61211 × 10⁻¹⁴ −2.11513 × 10⁻¹⁵ k = 6  1.35876 × 10⁻⁹ 4.88064 × 10⁻¹¹  5.67002 × 10⁻¹³  2.57805 × 10⁻¹⁵  3.29139 × 10⁻¹⁸ k =8 −7.02524 × 10⁻¹⁴ −1.36769 × 10⁻¹⁵ −7.18941 × 10⁻¹⁸ k = 10  9.65035 ×10⁻¹⁹ Bjk j = 6 j = 7 j = 8 j = 9 j = 10 k = 0  1.44673 × 10⁻⁹  1.23054× 10⁻¹¹  2.13590 × 10⁻¹⁴ −1.82326 × 10⁻¹⁶ −7.11582 × 10⁻¹⁹ k = 2−1.15121 × 10⁻¹³ −1.24718 × 10⁻¹⁵ −3.74717 × 10⁻¹⁸ k = 4 −8.26515 ×10⁻¹⁸

[0188] TABLE 9 Example 1 Reflecting Surface S5 N0 = N1 = 1 LocalCoordinates x y z Position 90.6219 0.0000 0.0000 Vector VX 1.0000 0.00000.0000 VY 0.0000 1.0000 0.0000 VZ 0.0000 0.0000 1.0000 C0 0.000000

[0189] TABLE 10 Example 1 Projection Surface S6 N0 = N1 = 1 LocalCoordinates x y z Position −109.3781 −547.8375 0.0000 Vector VX −1.00000.0000 0.0000 VY 0.0000 −1.0000 0.0000 VZ 0.0000 0.0000 1.0000 C00.000000

[0190] TABLE 11 Example 2 Display Surface S0 N0 = N1 = 1 LocalCoordinates x y z Position 0.0000 0.0000 0.0000 Vector VX 1.0000 0.00000.0000 VY 0.0000 1.0000 0.0000 VZ 0.0000 0.0000 1.0000 C0 0.000000

[0191] TABLE 12 Example 2 Reflecting Surface S1 N0 = N1 = 1 LocalCoordinates x y z Position 115.4561 −17.9518 0.0000 Vector VX 1.00000.0024 0.0000 VY −0.0024 1.0000 0.0000 VZ 0.0000 0.0000 1.0000 C0−0.007256 ε A4 A6 A8 A10 A12 1.0 −3.03358 × 10⁻⁹ −1.39052 × 10⁻¹²6.22246 × 10⁻¹⁶ −1.50263 × 10⁻¹⁹ 0.00000 × 10⁰

[0192] TABLE 13 Example 2 Pupil Plane APR N0 = N1 = 1 Local Coordinatesx y z Position 48.0000 −17.7500 0.0000 Vector VX −1.0000 0.0000 0.0000VY 0.0000 1.0000 0.0000 VZ 0.0000 0.0000 −1.0000 C0 0.000000 R 14.500000

[0193] TABLE 14 Example 2 Reflecting Surface S2 N0 = N1 = 1 LocalCoordinates x y z Position 4.7679 −14.4679 0.0000 Vector VX −0.9998−0.0206 0.0000 VY −0.0206 0.9998 0.0000 VZ 0.0000 0.0000 −1.0000 C00.006847 ε A4 A6 A8 A10 A12 1.0 4.61969 × 10⁻⁷ −5.13932 × 10⁻¹⁰ 1.06677× 10⁻¹² −1.04751 × 10⁻¹⁵ 4.52949 × 10⁻¹⁹

[0194] TABLE 15 Example 2 Reflecting Surface S3 N0 = N1 = 1 LocalCoordinates x y z Position 112.9764 −7.6544 0.0000 Vector VX 0.9995−0.0313 0.0000 VY 0.0313 0.9995 0.0000 VZ 0.0000 0.0000 1.0000 C00.007888 ε A4 A6 A8 A10 A12 1.0 3.40076 × 10⁻⁸ −2.52102 × 10⁻¹¹ −3.48433× 10⁻¹⁵ 1.13295 × 10⁻¹⁹ 0.00000 × 10⁰ Bjk j = 0 j = 1 j = 2 j = 3 j = 4j = 5 k = 0  0.00000 × 10⁰  0.00000 × 10⁰ −6.77427 × 10⁻³ −4.68604 ×10⁻⁵ −4.89629 × 10⁻⁷  1.29013 × 10⁻⁹ k = 2 −2.73509 × 10⁻³  5.65785 ×10⁻⁵ −3.30783 × 10⁻⁷ −2.18317 × 10⁻⁸ −3.01333 × 10⁻¹⁰ −5.30638 × 10⁻¹² k= 4 −6.35040 × 10⁻⁷ −3.94313 × 10⁻⁸ −1.00084 × 10⁻⁹ −2.18172 × 10⁻¹¹−3.02263 × 10⁻¹³ −2.62883 × 10⁻¹⁵ k = 6 −6.48372 × 10⁻¹⁰ −3.60308 ×10⁻¹¹ −6.03649 × 10⁻¹³ −4.89780 × 10⁻¹⁵ −1.65871 × 10⁻¹⁷ k = 8 −7.0674 ×10⁻¹⁴  2.28215 × 10⁻¹⁵  1.50126 × 10⁻¹⁷ k = 10  5.38948 × 10⁻¹⁷ Bjk j =6 j = 7 j = 8 j = 9 j = 10 k = 0  2.07728 × 10⁻¹¹ −1.18182 × 10⁻¹²−1.36299 × 10⁻¹⁴ −1.01150 × 10⁻¹⁶ −3.57759 × 10⁻¹⁹ k = 2 −5.02627 ×10⁻¹⁴ −4.50028 × 10⁻¹⁶ −1.91401 × 10⁻¹⁸ k = 4 −9.73344 × 10⁻¹⁸

[0195] TABLE 16 Example 2 Reflecting Surface S4 N0 = N1 = 1 LocalCoordinates x y z Position −36.5184 −5.1790 0.0000 Vector VX −0.92610.3774 0.0000 VY 0.3774 0.9261 0.0000 VZ 0.0000 0.0000 −1.0000 C0−0.005895 ε A4 A6 A8 A10 A12 1.0 −1.69630 × 10⁻⁶ 2.14389 × 10⁻¹⁰−1.30414 × 10⁻¹⁵ −1.08272 × 10⁻¹⁹ 3.51073 × 10⁻²⁴ Bjk j = 0 j = 1 j = 2j = 3 j = 4 j = 5 k = 0  0.00000 × 10⁰  0.00000 × 10⁻⁰ −4.33336 × 10⁻²−1.23194 × 10⁻³ −9.02325 × 10⁻⁶  4.56377 × 10⁻⁸ k = 2  6.90051 × 10⁻³−4.56827 × 10⁻⁴ −7.48434 × 10⁻⁶ −4.77783 × 10⁻⁸  1.08881 × 10⁻¹² 1.23589 × 10⁻¹² k = 4 −2.44526 × 10⁻⁶ −3.526 × 10⁻⁸  5.55124 × 10⁻¹⁰ 1.44621 × 10⁻¹¹ −8.61666 × 10⁻¹⁴ −2.11987 × 10⁻¹⁵ k = 6  1.37373 × 10⁻⁹ 4.93825 × 10⁻¹¹  5.65623 × 10⁻¹³  2.51280 × 10⁻¹⁵  3.10833 × 10⁻¹⁸ k =8 −6.l3527 × 10⁻¹⁴ −1.31218 × 10⁻¹⁵ −7.26368 × 10⁻¹⁸ k = 10  7.45150 ×10⁻¹⁹ Bjk j = 6 j = 7 j = 8 j = 9 j = 10 k = 0  1.44658 × 10⁻⁹  1.23067× 10⁻¹¹  2.13398 × 10⁻¹⁴ −1.82367 × 10⁻¹⁶ −7.10118 × 10⁻¹⁹ k = 2−1.15271 × 10⁻¹³ −1.24738 × 10⁻¹⁵ −3.73274 × 10⁻¹⁸ k = 4 −8.25915 ×10⁻¹⁸

[0196] TABLE 17 Example 2 Reflecting Surface S5 N0 = N1 = 1 LocalCoordinates x y z Position 124.2382 0.0000 0.0000 Vector VX 1.00000.0000 0.0000 VY 0.0000 1.0000 0.0000 VZ 0.0000 0.0000 1.0000 C00.000000

[0197] TABLE 18 Example 1 Projection Surface S6 N0 = N1 = 1 LocalCoordinates x y z Position −75.7618 −540.0081 0.0000 Vector VX −1.00000.0000 0.0000 VY 0.0000 −1.0000 0.0000 VZ 0.0000 0.0000 1.0000 C00.000000

[0198] TABLE 19 Example 3 Display Surface S0 N0 = N1 = 1 LocalCoordinates x y z Position 0.0000 0.0000 0.0000 Vector VX 1.0000 0.00000.0000 VY 0.0000 1.0000 0.0000 VZ 0.0000 0.0000 1.0000 C0 0.000000

[0199] TABLE 20 Example 3 Reflecting Surface S1 N0 = N1 = 1 LocalCoordinates x y z Position 101.6282 −16.6934 0.0000 Vector VX 0.9998−0.0195 0.0000 VY 0.0195 0.9998 0.0000 VZ 0.0000 0.0000 1.0000 C0−0.007814 ε A4 A6 A8 A10 A12 1.0 −4.90903 × 10⁻⁹ 1.95361 × 10⁻¹²−4.94963 × 10⁻¹⁵ 3.52855 × 10⁻¹⁸ 0.00000 × 10⁰

[0200] TABLE 21 Example 3 Pupil Plane APR N0 = N1 = 1 Local Coordinatesx y z Position 36.0000 −26.5000 0.0000 Vector VX −1.0000 0.0000 0.0000VY 0.0000 1.0000 0.0000 VZ 0.0000 0.0000 −1.0000 C0 0.000000 R 12.800000

[0201] TABLE 22 Example 3 Reflecting Surface S2 N0 = N1 = 1 LocalCoordinates x y z Position −6.9522 −12.2530 0.0000 Vector VX −0.99990.0106 0.0000 VY 0.0106 0.9999 0.0000 VZ 0.0000 0.0000 −1.0000 C00.006784 ε A4 A6 A8 A10 A12 1.0 4.94051 × 10⁻⁷ −1.33387 × 10⁻¹⁰ 1.43402× 10⁻¹³ 5.16304 × 10⁻¹⁷ −3.82708 × 10⁻²⁰

[0202] TABLE 23 Example 3 Reflecting Surface S3 N0 = N1 = 1 LocalCoordinates x y z Position 97.7147 −10.1564 0.0000 Vector VX 0.99970.0251 0.0000 VY −0.0251 0.9997 0.0000 VZ 0.0000 0.0000 1.0000 C00.008393 ε A4 A6 A8 A10 A12 1.0 −3.31188 × 10⁻⁸ −3.07605 × 10⁻¹¹−3.91714 × 10⁻¹⁵ 9.45955 × 10⁻²⁰ 0.00000 × 10⁰ Bjk j = 0 j = 1 j = 2 j =3 j = 4 j = 5 k = 0  0.00000 × 10⁰  0.00000 × 10⁰ −7.85975 × 10⁻³−5.59186 × 10⁻⁵ −5.77117 × 10⁻⁷ −5.12096 × 10⁻¹⁰ k = 2 −4.37752 × 10⁻³ 2.25556 × 10⁻⁵ −4.93592 × 10⁻⁷ −2.36270 × 10⁻⁸ −3.22127 × 10⁻¹⁰−5.54710 × 10⁻¹² k = 4 −4.43585 × 10⁻⁸ −2.03477 × 10⁻⁸ −9.41646 × 10⁻¹⁰−2.57965 × 10⁻¹¹ −3.33737 × 10⁻¹³ −2.58191 × 10⁻¹⁵ k = 6 −5.97881 ×10⁻¹⁰ −3.82987 × 10⁻¹¹ −7.98462 × 10⁻¹³ −7.46051 × 10⁻¹⁵ −2.63449 ×10⁻¹⁷ k = 8 −4.13400 × 10⁻¹⁴ −3.34889 × 10⁻¹⁶ −1.19825 × 10⁻¹⁸ k = 10 6.82597 × 10⁻¹⁸ Bjk j = 6 j = 7 j = 8 j = 9 j = 10 k = 0  1.34529 ×10⁻¹¹ −1.26912 × 10⁻¹² −1.42047 × 10⁻¹⁴ −1.08131 × 10⁻¹⁶ −3.70085 ×10⁻¹⁹ k = 2 −5.20931 × 10⁻¹⁴ −5.25768 × 10⁻¹⁶ −2.30273 × 10⁻¹⁸ k = 4−8.83935 × 10⁻¹⁸

[0203] TABLE 24 Example 3 Reflecting Surface S4 N0 = N1 = 1 LocalCoordinates x y z Position −49.4975 −0.7471 0.0000 Vector VX −0.93390.3576 0.0000 VY 0.3576 0.9339 0.0000 VZ 0.0000 0.0000 −1.0000 C0−0.005880 ε A4 A6 A8 A10 A12 1.0 −1.69296 × 10⁻⁶ 2.14551 × 10⁻¹⁰−1.31939 × 10⁻¹⁵ −1.07605 × 10⁻¹⁹ 3.47888 × 10⁻²⁴ Bjk j = 0 j = 1 j = 2j = 3 j = 4 j = 5 k = 0  0.00000 × 10⁰  0.00000 × 10⁰ −4.28053 × 10⁻²−1.22611 × 10⁻³ −9.01279 × 10⁻⁶  4.55202 × 10⁻⁸ k = 2  6.94508 × 10⁻³−4.59022 × 10⁻⁴ −7.53319 × 10⁻⁶ −4.80275 × 10⁻⁸  1.76668 × 10⁻¹² 1.24503 × 10⁻¹² k = 4 −2.36282 × 10⁻⁶ −3.18743 × 10⁻⁸  5.97328 × 10⁻¹⁰ 1.45260 × 10⁻¹¹ −8.80551 × 10⁻¹⁴ −2.12332 × 10⁻¹⁵ k = 6  1.38150 × 10⁻⁹ 4.92942 × 10⁻¹¹  5.59369 × 10⁻¹³  2.44035 × 10⁻¹⁵  2.82812 × 10⁻¹⁸ k =8 −6.44115 × 10⁻¹⁴ −1.37089 × 10⁻¹⁵ −7.33570 × 10⁻¹⁸ k = 10  3.26076 ×10⁻¹⁹ Bjk j = 6 j = 7 j = 8 j = 9 j = 10 k = 0  1.44670 × 10⁻⁹  1.23094× 10⁻¹¹  2.13073 × 10⁻¹⁴ −1.82245 × 10⁻¹⁶ −7.07774 × 10⁻¹⁹ k = 2−1.15306 × 10⁻¹³ −1.24783 × 10⁻¹⁵ −3.73197 × 10⁻¹⁸ k = 4 −8.21184 ×10⁻¹⁸

[0204] TABLE 25 Example 3 Reflecting Surface S5 N0 = N1 = 1 LocalCoordinates x y z Position 111.5073 0.0000 0.0000 Vector VX 1.00000.0000 0.0000 VY 0.0000 1.0000 0.0000 VZ 0.0000 0.0000 1.0000 C00.000000

[0205] TABLE 26 Example 3 Projection Surface S6 N0 = N1 = 1 LocalCoordinates x y z Position −88.4927 −532.1060 0.0000 Vector VX −1.00000.0000 0.0000 VY 0.0000 −1.0000 0.0000 VZ 0.0000 0.0000 1.0000 C00.000000

[0206] TABLE 27 Example 4 Display Surface S0 N0 = N1 = 1 LocalCoordinates x y z Position 0.0000 0.0000 0.0000 Vector VX 1.0000 0.00000.0000 VY 0.0000 1.0000 0.0000 VZ 0.0000 0.0000 1.0000 C0 0.000000

[0207] TABLE 28 Example 4 Pupil Plane APR N0 = N1 = 1 Local Coordinatesx y z Position 40.0000 −8.0000 0.0000 Vector VX 1.0000 0.0000 0.0000 VY0.0000 1.0000 0.0000 VZ 0.0000 0.0000 1.0000 C0 0.000000 R 6.750000

[0208] TABLE 29 Example 4 Reflecting Surface S1 N0 = N1 = 1 LocalCoordinates x y z Position 101.8029 −20.3753 0.0000 Vector VX 1.00000.0053 0.0000 VY −0.0053 1.0000 0.0000 VZ 0.0000 0.0000 1.0000 C0−0.012573 ε A4 A6 A8 A10 A12 1.0 3.93713 × 10⁻⁷ 1.49159 × 10⁻⁹ 9.71942 ×10⁻¹⁵ −4.32620 × 10⁻¹⁸ 0.00000 × 10⁰ Bjk j = 0 j = 1 j = 2 j = 3 j = 4 j= 5 k = 0  0.00000 × 10⁰  0.00000 × 10⁰  2.00884 × 10⁻³ 1.91578 × 10⁻⁶−2.40364 × 10⁻⁷ 2.20863 × 10⁻¹⁰ k = 2  1.59252 × 10⁻³  1.65260 × 10⁻⁶−5.04336 × 10⁻⁷ 4.76042 × 10⁻¹¹ −4.40812 × 10⁻⁹ 0.00000 × 10⁰ k = 4−2.70902 × 10⁻⁷ −3.30009 × 10⁻¹⁰ −4.42708 × 10⁻⁹ 0.00000 × 10⁰  0.00000× 10⁰ 0.00000 × 10⁰ k = 6 −1.47575 × 10⁻⁹  0.00000 × 10⁰  0.00000 × 10⁰0.00000 × 10⁰  0.00000 × 10⁰ k = 8  0.00000 × 10⁰  0.00000 × 10⁰ 0.00000 × 10⁰ k = 10  0.00000 × 10⁰ Bjk j = 6 j = 7 j = 8 j = 9 j = 10k = 0 −1.47127 × 10⁻⁹ 0.00000 × 10⁰ 0.00000 × 10⁰ 0.00000 × 10⁰ 0.00000× 10⁰ k = 2  0.00000 × 10⁰ 0.00000 × 10⁰ 0.00000 × 10⁰ k = 4  0.00000 ×10⁰

[0209] TABLE 30 Example 4 Reflecting Surface S2 N0 = N1 = 1 LocalCoordinates x y z Position 12.5841 −39.2297 0.0000 Vector VX −0.99520.0978 0.0000 VY 0.0978 0.9952 0.0000 VZ 0.0000 0.0000 −1.0000 C00.044024 ε A4 A6 A8 A10 A12 1.0 −4.87501 × 10⁻⁴ −4.32896 × 10⁻⁶ −6.34246× 10⁻¹¹ 1.11218 × 10⁻¹³ −2.01234 × 10⁻¹⁶ Bjk j = 0 j = 1 j = 2 j = 3 j =4 j = 5 k = 0  0.00000 × 10⁰  0.00000 × 10⁰ −1.42523 × 10⁻² −1.32737 ×10⁻⁴ 4.81629 × 10⁻⁴ −3.44610 × 10⁻⁷ k = 2 −9.49763 × 10⁻³ −2.41389 ×10⁻⁴  9.64792 × 10⁻⁴ −9.61107 × 10⁻⁷ 1.30127 × 10⁻⁵  0.00000 × 10⁰ k = 4 4.84935 × 10⁻⁴ −6.98502 × 10⁻⁷  1.30266 × 10⁻⁵  0.00000 × 10⁰ 0.00000 ×10⁰  0.00000 × 10⁰ k = 6  4.34057 × 10⁻⁶  0.00000 × 10⁰  0.00000 × 10⁰ 0.00000 × 10⁰ 0.00000 × 10⁰ k = 8  0.00000 × 10⁰  0.00000 × 10⁰ 0.00000 × 10⁰ k = 10  0.00000 × 10⁰ Bjk j = 6 j = 7 j = 8 j = 9 j = 10k = 0 4.33854 × 10⁻⁶ 0.00000 × 10⁰ 0.00000 × 10⁰ 0.00000 × 10⁰ 0.00000 ×10⁰ k = 2 0.00000 × 10⁰ 0.00000 × 10⁰ 0.00000 × 10⁰ k = 4 0.00000 × 10⁰

[0210] TABLE 31 Example 4 Reflecting Surface S3 N0 = N1 = 1 LocalCoordinates x y z Position 101.8026 −77.3632 0.0000 Vector VX 0.9951−0.0986 0.0000 VY 0.0986 0.9951 0.0000 VZ 0.0000 0.0000 1.0000 C00.008620 ε A4 A6 A8 A10 A12 1.0 2.14466 × 10⁻⁷ −4.67228 × 10⁻¹¹ −4.63566× 10⁻¹⁵ 4.77752 × 10⁻¹⁸ 0.00000 × 10⁰ Bjk j = 0 j = 1 j = 2 j = 3 j = 4j = 5 k = 0  0.00000 × 10⁰  0.00000 × 10⁰ −6.61267 × 10⁻³ −6.62783 ×10⁻⁶ −3.40700 × 10⁻⁷  3.39988 × 10⁻⁹ k = 2 −6.95327 × 10⁻³ −1.54197 ×10⁻⁶ −5.76526 × 10⁻⁷  5.19801 × 10⁻⁹  1.74131 × 10⁻¹⁰ −1.62011 × 10⁻¹² k= 4 −2.81046 × 10⁻⁷  1.87703 × 10⁻⁹  1.34552 × 10⁻¹⁰ −6.69182 × 10⁻¹³ 2.12931 × 10⁻¹⁴  2.58344 × 10⁻¹⁶ k = 6  3.99774 × 10⁻¹¹ −2.38038 ×10⁻¹³  1.84027 × 10⁻¹⁴  5.42145 × 10⁻¹⁸ −4.90278 × 10⁻¹⁷ k = 8  4.38169× 10⁻¹⁵  1.52645 × 10⁻¹⁷ −2.38690 × 10⁻¹⁷ k = 10 −4.74121 × 10⁻¹⁸ Bjk j= 6 j = 7 j = 8 j = 9 j = 10 k = 0  5.21016 × 10⁻¹¹ −8.25123 × 10⁻¹³−2.45323 × 10⁻¹⁵ −2.47599 × 10⁻¹⁸ −2.65756 × 10⁻¹⁸ k = 2  1.72914 ×10⁻¹⁴  5.21144 × 10⁻¹⁶ −2.41525 × 10⁻¹⁷ k = 4 −4.16200 × 10⁻¹⁷

[0211] TABLE 32 Example 4 Reflecting Surface S4 N0 = N1 = 1 LocalCoordinates x y z Position 6.3346 −97.3170 0.0000 Vector VX −0.90850.4179 0.0000 VY 0.4179 0.9085 0.0000 VZ 0.0000 0.0000 −1.0000 C0−0.001672 ε A4 A6 A8 A10 A12 1.0 1.93910 × 10⁻⁷ 4.40895 × 10⁻¹¹ −4.60538× 10⁻¹⁵ −6.66318 × 10⁻¹⁸ −1.04921 × 10⁻²² Bjk j = 0 j = 1 j = 2 j = 3 j= 4 j = 5 k = 0  0.00000 × 10⁰  0.00000 × 10⁰  3.63408 × 10⁻³  5.60033 ×10⁻⁵  1.09617 × 10⁻⁶  4.09506 × 10⁻⁸ k = 2  7.85586 × 10⁻³  1.30899 ×10⁻⁴  2.58152 × 10⁻⁶  5.89453 × 10⁻⁸  8.20095 × 10⁻¹⁰ −2.47675 × 10⁻¹² k= 4 −3.89420 × 10⁻⁷ −1.63610 × 10⁻⁸ −1.04482 × 10⁻⁹ −3.48702 × 10⁻¹¹−6.89304 × 10⁻¹³ −1.05752 × 10⁻¹⁴ k = 6 −4.95414 × 10⁻¹¹  3.81251 ×10⁻¹⁴  1.25618 × 10⁻¹³  6.66734 × 10⁻¹⁵  1.79935 × 10⁻¹⁶ k = 8  3.06365× 10⁻¹⁵  1.70276 × 10⁻¹⁶  3.83915 × 10⁻¹⁷ k = 10  8.08880 × 10⁻¹⁸ Bjk j= 6 j = 7 j = 8 j = 9 j = 10 k = 0  9.02998 × 10⁻¹⁰ −4.18456 × 10⁻¹²−4.82726 × 10⁻¹³ −6.37146 × 10⁻¹⁵ −2.80979 × 10⁻¹⁷ k = 2 −7.18032 ×10⁻¹³ −1.31925 × 10⁻¹⁴  9.14530 × 10⁻¹⁸ k = 4 −5.63936 × 10⁻¹⁷

[0212] TABLE 33 Example 4 Reflecting Surface S5 N0 = N1 = 1 LocalCoordinates x y z Position 101.8029 0.0000 0.0000 Vector VX 1.00000.0000 0.0000 VY 0.0000 1.0000 0.0000 VZ 0.0000 0.0000 1.0000 C00.000000

[0213] TABLE 34 Example 4 Projection Surface S6 N0 = N1 = 1 LocalCoordinates x y z Position −8.1971 −472.0624 0.0000 Vector VX −1.00000.0000 0.0000 VY 0.0000 −1.0000 0.0000 VZ 0.0000 0.0000 1.0000 C00.000000

[0214] TABLE 35 Example 5 Display Surface S0 N0 = N1 = 1 LocalCoordinates x y z Position 0.0000 0.0000 0.0000 Vector VX 1.0000 0.00000.0000 VY 0.0000 1.0000 0.0000 VZ 0.0000 0.0000 1.0000 C0 0.000000

[0215] TABLE 36 Example 5 Reflecting Surface S1 N0 = N1 = 1 LocalCoordinates x y z Position 63.5704 −15.8836 0.0000 Vector VX 0.9989−0.0460 0.0000 VY 0.0460 0.9989 0.0000 VZ 0.0000 0.0000 1.0000 C0−0.010935 ε A4 A6 A8 A10 A12 1.0 1.25724 × 10⁻⁵ 1.86170 × 10⁻⁷ −4.05135× 10⁻¹² 3.01351 × 10⁻¹⁴ 0.00000 × 10⁰ Bjk j = 0 j = 1 j = 2 j = 3 j = 4j = 5 k = 0  0.00000 × 10⁰  0.00000 × 10⁰ −4.43584 × 10⁻⁴ 2.31983 × 10⁻⁶−1.26143 × 10⁻⁵ 1.23098 × 10⁻⁹ k = 2 −1.34491 × 10⁻³  1.89361 × 10⁻⁶−2.53874 × 10⁻⁵ 1.79014 × 10⁻⁹ −5.57511 × 10⁻⁷ 0.00000 × 10⁰ k = 4−1.27249 × 10⁻⁵ −7.53181 × 10⁻¹⁰ −5.57968 × 10⁻⁷ 0.00000 × 10⁰  0.00000× 10⁰ 0.00000 × 10⁰ k = 6 −1.86192 × 10⁻⁷  0.00000 × 10⁰  0.00000 × 10⁰0.00000 × 10⁰  0.00000 × 10⁰ k = 8  0.00000 × 10⁰  0.00000 × 10⁰ 0.00000 × 10⁰ k = 10  0.00000 × 10⁰ Bjk j = 6 j = 7 j = 8 j = 9 j = 10k = 0 −1.86105 × 10⁻⁷ 0.00000 × 10⁰ 0.00000 × 10⁰ 0.00000 × 10⁰ 0.00000× 10⁰ k = 2  0.00000 × 10⁰ 0.00000 × 10⁰ 0.00000 × 10⁰ k = 4  0.00000 ×10⁰

[0216] TABLE 37 Example 5 Pupil Plane APR N0 = N1 = 1 Local Coordinatesx y z Position 50.0000 −17.0000 0.0000 Vector VX −1.0000 0.0000 0.0000VY 0.0000 1.0000 0.0000 VZ 0.0000 0.0000 −1.0000 C0 0.000000 R 8.700000

[0217] TABLE 38 Example 5 Reflecting Surface S2 N0 = N1 = 1 LocalCoordinates x y z Position −1.4357 −25.9385 0.0000 Vector VX −0.99260.1216 0.0000 VY 0.1216 0.9926 0.0000 VZ 0.0000 0.0000 −1.0000 C00.043194 ε A4 A6 A8 A10 A12 1.0 −4.52966 × 10⁻⁴ −4.18988 × 10⁻⁶ 7.40925× 10⁻¹¹ −7.60897 × 10⁻¹³ 1.94262 × 10⁻¹⁵ Bjk j = 0 j = 1 j = 2 j = 3 j =4 j = 5 k = 0  0.00000 × 10⁰  0.00000 × 10⁰ −1.66596 × 10⁻² −9.46800 ×10⁻⁵ 4.45974 × 10⁻⁴ −1.35981 × 10⁻⁷ k = 2 −9.69251 × 10⁻³ −2.44519 ×10⁻⁴  8.96345 × 10⁻⁴ −5.70625 × 10⁻⁷ 1.25560 × 10⁻⁵  0.00000 × 10⁰ k = 4 4.55105 × 10⁻⁴ −4.81406 × 10⁻⁷  1.25666 × 10⁻⁵  0.00000 × 10⁰ 0.00000 ×10⁰  0.00000 × 10⁰ k = 6  4.18834 × 10⁻⁶  0.00000 × 10⁰  0.00000 × 10⁰ 0.00000 × 10⁰ 0.00000 × 10⁰ k = 8  0.00000 × 10⁰  0.00000 × 10⁰ 0.00000 × 10⁰ k = 10  0.00000 × 10⁰ Bjk j = 6 j = 7 j = 8 j = 9 j = 10k = 0 4.17880 × 10⁻⁶ 0.00000 × 10⁰ 0.00000 × 10⁰ 0.00000 × 10⁰ 0.00000 ×10⁰ k = 2 0.00000 × 10⁰ 0.00000 × 10⁰ 0.00000 × 10⁰ k = 4 0.00000 × 10⁰

[0218] TABLE 39 Example 5 Reflecting Surface S3 N0 = N1 = 1 LocalCoordinates x y z Position 100.3365 −68.4122 0.0000 Vector VX 0.99250.1224 0.0000 VY −0.1224 0.9925 0.0000 VZ 0.0000 0.0000 1.0000 C00.006364 ε A4 A6 A8 A10 A12 1.0 2.28056 × 10⁻⁷ −4.57134 × 10⁻¹¹ −4.50292× 10⁻¹⁵ 4.78237 × 10⁻¹⁸ 0.00000 × 10⁰ Bjk j = 0 j = 1 j = 2 j = 3 j = 4j = 5 k = 0  0.00000 × 10⁰  0.00000 × 10⁰ −3.89379 × 10⁻³ −6.30657 ×10⁻⁶ −3.77568 × 10⁻⁷  5.53503 × 10⁻¹⁰ k = 2 −4.45623 × 10⁻³  9.38273 ×10⁻⁶ −4.65719 × 10⁻⁷  2.03768 × 10⁻⁹  1.63626 × 10⁻¹⁰ −3.30206 × 10⁻¹⁴ k= 4 −3.03236 × 10⁻⁷ −5.64446 × 10⁻¹⁰  1.27401 × 10⁻¹⁰ −6.31578 × 10⁻¹⁴ 2.56344 × 10⁻¹⁴  7.08908 × 10⁻¹⁷ k = 6  4.54551 × 10⁻¹¹ −1.90082 ×10⁻¹⁴  1.86998 × 10⁻¹⁴ −4.24892 × 10⁻¹⁸ −4.78662 × 10⁻¹⁷ k = 8  4.45436× 10⁻¹⁵  5.48344 × 10⁻¹⁸ −2.39022 × 10⁻¹⁷ k = 10 −4.77522 × 10⁻¹⁸ Bjk j= 6 j = 7 j = 8 j = 9 j = 10 k = 0  6.99868 × 10⁻¹¹  8.59898 × 10⁻¹³ 4.36970 × 10⁻¹⁴ 1.30934 × 10⁻¹⁵ 1.05237 × 10⁻¹⁷ k = 2  1.85583 × 10⁻¹⁴−2.97697 × 10⁻¹⁶ −2.94138 × 10⁻¹⁷ k = 4 −4.68875 × 10⁻¹⁷

[0219] TABLE 40 Example 5 Reflecting Surface S4 N0 = N1 = 1 LocalCoordinates x y z Position 19.9467 −128.3830 0.0000 Vector VX −0.97980.2000 0.0000 VY 0.2000 0.9798 0.0000 VZ 0.0000 0.0000 −1.0000 C0−0.004794 ε A4 A6 A8 A10 A12 1.0 −1.81730 × 10⁻⁸ 6.89428 × 10⁻¹¹ 5.19899× 10⁻¹⁵ −7.40999 × 10⁻¹⁸ −2.99449 × 10⁻²⁵ Bjk j = 0 j = 1 j = 2 j = 3 j= 4 j = 5 k = 0  0.00000 × 10⁰  0.00000 × 10⁰  3.09395 × 10⁻³  1.11583 ×10⁻⁵  1.03114 × 10⁻⁷  5.43111 × 10⁻⁹ k = 2  5.05181 × 10⁻³  2.55058 ×10⁻⁵  4.10929 × 10⁻⁷  4.51142 × 10⁻⁹ −1.86926 × 10⁻¹⁰ −7.19401 × 10⁻¹³ k= 4 −3.04900 × 10⁻⁹ −1.36475 × 10⁻⁹ −2.42120 × 10⁻¹⁰ −8.74626 × 10⁻¹³−3.84016 × 10⁻¹⁴  1.77316 × 10⁻¹⁷ k = 6 −6.57427 × 10⁻¹¹  5.94997 ×10⁻¹⁴ −1.98078 × 10⁻¹⁴  4.55095 × 10⁻¹⁷  7.45904 × 10⁻¹⁷ k = 8 −5.36094× 10⁻¹⁵ −2.64065 × 10⁻¹⁸  3.70927 × 10⁻¹⁷ k = 10  7.41820 × 10⁻¹⁸ Bjk j= 6 j = 7 j = 8 j = 9 j = 10 k = 0  1.88009 × 10⁻¹⁰  1.91127 × 10⁻¹²−8.48082 × 10⁻¹⁴ −1.57209 × 10⁻¹⁵ −7.22437 × 10⁻¹⁹ k = 2 −3.38507 ×10⁻¹⁴ −7.93454 × 10⁻¹⁷  3.69191 × 10⁻¹⁷ k = 4  7.44125 × 10⁻¹⁷

[0220] TABLE 41 Example 5 Reflecting Surface S5 N0 = N1 = 1 LocalCoordinates x y z Position 103.5699 0.0000 0.0000 Vector VX 1.00000.0000 0.0000 VY 0.0000 1.0000 0.0000 VZ 0.0000 0.0000 1.0000 C00.000000

[0221] TABLE 42 Example 5 Projection Surface S6 N0 = N1 = 1 LocalCoordinates x y z Position −16.4301 −478.3923 0.0000 Vector VX −1.00000.0000 0.0000 VY 0.0000 −1.0000 0.0000 VZ 0.0000 0.0000 1.0000 C00.000000

[0222] TABLE 43 Example 6 Display Surface S0 N0 = N1 = 1 LocalCoordinates x y z Position 0.0000 0.0000 0.0000 Vector VX 1.0000 0.00000.0000 VY 0.0000 1.0000 0.0000 VZ 0.0000 0.0000 1.0000 C0 0.000000

[0223] TABLE 44 Example 6 Reflecting Surface S1 N0 = N1 = 1 LocalCoordinates x y z Position 62.4807 −11.1468 0.0000 Vector VX 0.9999−0.0152 0.0000 VY 0.0152 0.9999 0.0000 VZ 0.0000 0.0000 1.0000 C0−0.008352 ε A4 A6 A8 A10 A12 1.0 1.95222 × 10⁻⁵ 1.95773 × 10⁻⁷ −3.59070× 10⁻¹³ 3.89395 × 10⁻¹⁶ 0.00000 × 10⁰ Bjk j = 0 j = 1 j = 2 j = 3 j = 4j = 5 k = 0  0.00000 × 10⁰  0.00000 × 10⁰ −2.00942 × 10⁻³ 2.13947 × 10⁻⁶−1.97120 × 10⁻⁵ 1.03253 × 10⁻⁹ k = 2 −2.74651 × 10⁻³  1.56985 × 10⁻⁶−3.94978 × 10⁻⁵ 7.56232 × 10⁻¹⁰ −5.86884 × 10⁻⁷ 0.00000 × 10⁰ k = 4−1.97985 × 10⁻⁵ −1.18174 × 10⁻⁹ −5.87120 × 10⁻⁷ 0.00000 × 10⁰  0.00000 ×10⁰ 0.00000 × 10⁰ k = 6 −1.95698 × 10⁻⁷  0.00000 × 10⁰  0.00000 × 10⁰0.00000 × 10⁰  0.00000 × 10⁰ k = 8  0.00000 × 10⁰  0.00000 × 10⁰ 0.00000 × 10⁰ k = 10  0.00000 × 10⁰ Bjk j = 6 j = 7 j = 8 j = 9 j = 10k = 0 −1.95633 × 10⁻⁷ 0.00000 × 10⁰ 0.00000 × 10⁰ 0.00000 × 10⁰ 0.00000× 10⁰ k = 2  0.00000 × 10⁰ 0.00000 × 10⁰ 0.00000 × 10⁰ k = 4  0.00000 ×10⁰

[0224] TABLE 45 Example 6 Pupil Plane APR N0 = N1 = 1 Local Coordinatesx y z Position 24.0000 −17.0000 0.0000 Vector VX −1.0000 0.0000 0.0000VY 0.0000 1.0000 0.0000 VZ 0.0000 0.0000 −1.0000 C0 0.000000 R 6.250000

[0225] TABLE 46 Example 6 Reflecting Surface S2 N0 = N1 = 1 LocalCoordinates x y z Position −2.5314 −20.8218 0.0000 Vector VX −0.99340.1144 0.0000 VY 0.1144 0.9934 0.0000 VZ 0.0000 0.0000 −1.0000 C00.027834 ε A4 A6 A8 A10 A12 1.0 −1.79016 × 10⁻⁴ −1.67583 × 10⁻⁶ 1.28774× 10⁻¹⁰ −1.95483 × 10⁻¹² 7.65360 × 10⁻¹⁵ Bjk j = 0 j = 1 j = 2 j = 3 j =4 j = 5 k = 0  0.00000 × 10⁰  0.00000 × 10⁰ −7.66855 × 10⁻³ −1.05589 ×10⁴ 1.80094 × 10⁻⁴ −2.20048 × 10⁻⁷ k = 2 −1.34309 × 10⁻³ −2.47669 × 10⁴ 3.64061 × 10⁻⁴ −6.77025 × 10⁻⁷ 5.06258 × 10⁻⁶  0.00000 × 10⁰ k = 4 1.89105 × 10⁻⁴ −4.18413 × 10⁻⁷  5.07642 × 10⁻⁶  0.00000 × 10⁰ 0.00000 ×10⁰  0.00000 × 10⁰ k = 6  1.69143 × 10⁻⁶  0.00000 × 10⁰  0.00000 × 10⁰ 0.00000 × 10⁰ 0.00000 × 10⁰ k = 8  0.00000 × 10⁰  0.00000 × 10⁰ 0.00000 × 10⁰ k = 10  0.00000 × 10⁰ Bjk j = 6 j = 7 j = 8 j = 9 j = 10k = 0 1.68175 × 10⁻⁶ 0.00000 × 10⁰ 0.00000 × 10⁰ 0.00000 × 10⁰ 0.00000 ×10⁰ k = 2 0.00000 × 10⁰ 0.00000 × 10⁰ 0.00000 × 10⁰ k = 4 0.00000 × 10⁰

[0226] TABLE 47 Example 6 Reflecting Surface S3 N0 = N1 = 1 LocalCoordinates x y z Position 120.1083 −68.6501 0.0000 Vector VX 0.99460.1034 0.0000 VY −0.1034 0.9946 0.0000 VZ 0.0000 0.0000 1.0000 C00.004000 ε A4 A6 A8 A10 A12 1.0 2.42154 × 10⁻⁷ −4.54612 × 10⁻¹¹ −4.53196× 10⁻¹⁵ 4.78069 × 10⁻¹⁸ 0.00000 × 10⁰ Bjk j = 0 j = 1 j = 2 j = 3 j = 4j = 5 k = 0  0.00000 × 10⁰  0.00000 × 10⁰ −3.19555 × 10⁻³ −4.89598 ×10⁻⁶ −3.50217 × 10⁻⁷ −6.21325 × 10⁻¹⁰ k = 2 −3.47325 × 10⁻³  3.83920 ×10⁻⁶ −4.65460 × 10⁻⁷  1.42488 × 10⁻⁹  1.44970 × 10⁻¹⁰ −1.49391 × 10⁻¹³ k= 4 −2.68538 × 10⁻⁷ −6.36563 × 10⁻¹¹  1.32216 × 10⁻¹⁰ −1.10135 × 10⁻¹³ 2.68177 × 10⁻¹⁴  3.41538 × 10⁻¹⁷ k = 6  4.52265 × 10⁻¹¹ −1.11849 ×10⁻¹⁴  1.82407 × 10⁻¹⁴  1.71914 × 10⁻¹⁸ −4.77890 × 10⁻¹⁷ k = 8  4.51594× 10⁻¹⁵  7.81835 × 10⁻¹⁹ −2.39021 × 10⁻¹⁷ k = 10 −4.77948 × 10⁻¹⁸ Bjk j= 6 j = 7 j = 8 j = 9 j = 10 k = 0  9.66229 × 10⁻¹¹  1.21715 × 10⁻¹²−2.21158 × 10⁻¹⁶ −2.25333 × 10⁻¹⁶ −5.45701 × 10⁻¹⁸ k = 2  1.59084 ×10⁻¹⁴ −8.78533 × 10⁻¹⁷ −2.49975 × 10⁻¹⁷ k = 4 −4.74824 × 10⁻¹⁷

[0227] TABLE 48 Example 6 Reflecting Surface S4 N0 = N1 = 1 LocalCoordinates x y z Position 31.6335 −126.5920 0.0000 Vector VX −0.96350.2677 0.0000 VY 0.2677 0.9635 0.0000 VZ 0.0000 0.0000 −1.0000 C0−0.004044 ε A4 A6 A8 A10 A12 1.0 −1.46748 × 10⁻⁸ 6.91335 × 10⁻¹¹ 5.22720× 10⁻¹⁵ −7.41421 × 10⁻¹⁸ 0.00000 × 10⁰ Bjk j = 0 j = 1 j = 2 j = 3 j = 4j = 5 k = 0  0.00000 × 10⁰  0.00000 × 10⁰  2.83030 × 10⁻³  9.52598 ×10⁻⁶  1.08858 × 10⁻⁷  4.29336 × 10⁻⁹ k = 2  5.08913 × 10⁻³  2.79153 ×10⁻⁵  3.96142 × 10⁻⁷  4.57770 × 10⁻⁹ −1.53563 × 10⁻¹⁰  6.92191 × 10⁻¹⁴ k= 4 −3.26504 × 10⁻⁹ −1.04841 × 10⁻⁹ −2.34139 × 10⁻¹⁰ −6.24686 × 10⁻¹³−3.88274 × 10⁻¹⁴ −4.94681 × 10⁻¹⁷ k = 6 −6.75734 × 10⁻¹¹  1.67018 ×10⁻¹⁴ −2.01313 × 10⁻¹⁴  2.54704 × 10⁻¹⁷  7.43461 × 10⁻¹⁷ k = 8 −5.29126× 10⁻¹⁵ −1.89384 × 10⁻¹⁹  3.70810 × 10⁻¹⁷ k = 10  7.41814 × 10⁻¹⁸ Bjk j= 6 j = 7 j = 8 j = 9 j = 10 k = 0  8.77554 × 10⁻¹¹  1.03950 × 10⁻¹²−4.68539 × 10⁻¹⁴ −7.83833 × 10⁻¹⁶ 3.50804 × 10⁻¹⁸ k = 2 −3.08646 × 10⁻¹⁴−1.39727 × 10⁻¹⁶  3.64635 × 10⁻¹⁷ k = 4  7.39562 × 10⁻¹⁷

[0228] TABLE 49 Example 6 Reflecting Surface S5 N0 = N1 = 1 LocalCoordinates x y z Position 127.4805 0.0000 0.0000 Vector VX 1.00000.0000 0.0000 VY 0.0000 1.0000 0.0000 VZ 0.0000 0.0000 1.0000 C00.000000

[0229] TABLE 50 Example 6 Projection Surface S6 N0 = N1 = 1 LocalCoordinates x y z Position −17.5195 −626.5924 0.0000 Vector VX −1.00000.0000 0.0000 VY 0.0000 −1.0000 0.0000 VZ 0.0000 0.0000 1.0000 C00.000000

[0230] TABLE 51 Example 7 Display Surface S0 N0 = N1 = 1 LocalCoordinates x y z Position 0.0000 0.0000 0.0000 Vector VX 1.0000 0.00000.0000 VY 0.0000 1.0000 0.0000 VZ 0.0000 0.0000 1.0000 C0 0.000000

[0231] TABLE 52 Example 7 Reflecting Surface S1 N0 = N1 = 1 LocalCoordinates x y z Position 75.4182 13.4971 0.0000 Vector VX 0.94750.3196 0.0000 VY −0.3196 0.9475 0.0000 VZ 0.0000 0.0000 1.0000 C0−0.017525 ε A4 A6 A8 A10 A12 1.0 −1.66382 × 10⁻⁷ 2.82609 × 10⁻¹⁰ 4.81274× 10⁻¹⁵ 1.10802 × 10⁻¹⁷ 0.00000 × 10⁰ Bjk j = 0 j = 1 j = 2 j = 3 j = 4j = 5 k = 0  0.00000 × 10⁰  0.00000 × 10⁰  3.73857 × 10⁻³ −5.01341 ×10⁻⁷  6.80585 × 10⁻⁷ 2.63039 × 10⁻¹¹ k = 2  3.21110 × 10⁻³ −1.35376 ×10⁻⁶  1.40081 × 10⁻⁶  3.23581 × 10⁻⁹ −4.59668 × 10⁻¹⁰ 0.00000 × 10⁰ k =4  6.55625 × 10⁻⁷  1.80319 × 10⁻¹⁰ −5.12242 × 10⁻¹⁰  0.00000 × 10⁰ 0.00000 × 10⁰ 0.00000 × 10⁰ k = 6 −1.79598 × 10⁻¹⁰  0.00000 × 10⁰ 0.00000 × 10⁰  0.00000 × 10⁰  0.00000 × 10⁰ k = 8  0.00000 × 10⁰ 0.00000 × 10⁰  0.00000 × 10⁰ k = 10  0.00000 × 10⁰ Bjk j = 6 j = 7 j =8 j = 9 j = 10 k = 0 −1.67688 × 10⁻¹⁰ 0.00000 × 10⁰ 0.00000 × 10⁰0.00000 × 10⁰ 0.00000 × 10⁰ k = 2  0.00000 × 10⁰ 0.00000 × 10⁰ 0.00000 ×10⁰ k = 4  0.00000 × 10⁰

[0232] TABLE 53 Example 7 Pupil Plane APR N0 = N1 = 1 Local Coordinatesx y z Position 32.6886 −17.5000 0.0000 Vector VX −1.0000 0.0000 0.0000VY 0.0000 1.0000 0.0000 VZ 0.0000 0.0000 −1.0000 C0 0.000000 R 9.500000

[0233] TABLE 54 Example 7 Reflecting Surface S2 N0 = N1 = 1 LocalCoordinates x y z Position 3.2516 −26.2261 0.0000 Vector VX −1.0000−0.0021 0.0000 VY −0.0021 1.0000 0.0000 VZ 0.0000 0.0000 −1.0000 C00.029943 ε A4 A6 A8 A10 A12 1.0 −6.67190 × 10⁻⁴ −5.76813 × 10⁻⁶ 8.17928× 10⁻¹¹ −3.16719 × 10⁻¹³ 3.23298 × 10⁻¹⁶ Bjk j = 0 j = 1 j = 2 j = 3 j =4 j = 5 k = 0  0.00000 × 10⁰  0.00000 × 10⁰ −1.01265 × 10⁻² −5.73887 ×10⁻⁵ 6.65231 × 10⁻⁴ −5.43254 × 10⁻⁸ k = 2 −4.67729 × 10⁻³ −8.88926 ×10⁻⁵  1.33180 × 10⁻³ −2.21354 × 10⁻⁷ 1.72808 × 10⁻⁵  0.00000 × 10⁰ k = 4 6.68079 × 10⁻⁴ −1.51675 × 10⁻⁷  1.72872 × 10⁻⁵  0.00000 × 10⁰ 0.00000 ×10⁰  0.00000 × 10⁰ k = 6  5.76484 × 10⁻⁶  0.00000 × 10⁰  0.00000 × 10⁰ 0.00000 × 10⁰ 0.00000 × 10⁰ k = 8  0.00000 × 10⁰  0.00000 × 10⁰ 0.00000 × 10⁰ k = 10  0.00000 × 10⁰ Bjk j = 6 j = 7 j = 8 j = 9 j = 10k = 0 5.75950 × 10⁻⁶ 0.00000 × 10⁰ 0.00000 × 10⁰ 0.00000 × 10⁰ 0.00000 ×10⁰ k = 2 0.00000 × 10⁰ 0.00000 × 10⁰ 0.00000 × 10⁰ k = 4 0.00000 × 10⁰

[0234] TABLE 55 Example 7 Reflecting Surface S3 N0 = N1 = 1 LocalCoordinates x y z Position 82.8882 −23.7811 0.0000 Vector VX 0.9984−0.0558 0.0000 VY 0.0558 0.9984 0.0000 VZ 0.0000 0.0000 1.0000 C00.008884 ε A4 A6 A8 A10 A12 1.0 6.42863 × 10⁻⁷ −1.57431 × 10⁻¹⁰ −3.38299× 10⁻¹⁴ 2.89096 × 10⁻¹⁸ 0.00000 × 10⁰ Bjk j = 0 j = 1 j = 2 j = 3 j = 4j = 5 k = 0  0.00000 × 10⁰  0.00000 × 10⁰ −3.97653 × 10⁻³  8.09710 ×10⁻⁵  7.40631 × 10⁻⁷ −4.93872 × 10⁻⁹ k = 2 −6.38754 × 10⁻³  5.32223 ×10⁻⁵  9.56721 × 10⁻⁷  3.69191 × 10⁻⁸ −3.55872 × 10⁻¹⁰ −4.62424 × 10⁻¹¹ k= 4 −6.26197 × 10⁻⁷ −7.40298 × 10⁻⁹ −5.66599 × 10⁻¹⁰ −4.49197 × 10⁻¹¹−7.08572 × 10⁻¹³ −8.61053 × 10⁻¹⁵ k = 6  1.00306 × 10⁻¹⁰ −6.82284 ×10⁻¹² −1.45392 × 10⁻¹³ −4.51260 × 10⁻¹⁵ −5.41853 × 10⁻¹⁷ k = 8  2.74914× 10⁻¹⁴ −2.83504 × 10⁻¹⁶ −1.72951 × 10⁻¹⁷ k = 10 −3.15292 × 10⁻¹⁸ Bjk j= 6 j = 7 j = 8 j = 9 j = 10 k = 0 −6.96493 × 10⁻¹⁰ −1.86322 × 10⁻¹¹−1.57288 × 10⁻¹³ −8.97245 × 10⁻¹⁶ −4.20010 × 10⁻¹⁸ k = 2 −6.99855 ×10⁻¹³ −6.99205 × 10⁻¹⁵ −3.75748 × 10⁻¹⁷ k = 4 −5.95656 × 10⁻¹⁷

[0235] TABLE 56 Example 7 Reflecting Surface S4 N0 = N1 = 1 LocalCoordinates x y z Position −32.1140 6.8008 0.0000 Vector VX −0.98730.1586 0.0000 VY 0.1586 0.9873 0.0000 VZ 0.0000 0.0000 −1.0000 C0−0.004801 ε A4 A6 A8 A10 A12 1.0 −1.75892 × 10⁻⁶ 2.16761 × 10⁻¹⁰−1.41979 × 10⁻¹⁵ −1.02102 × 10⁻¹⁹ 3.17334 × 10⁻²⁴ Bjk j = 0 j = 1 j = 2j = 3 j = 4 j = 5 k = 0  0.00000 × 10⁰  0.00000 × 10⁰ −3.85943 × 10⁻²−1.19159 × 10⁻³ −8.95224 × 10⁻⁶  4.47919 × 10⁻⁸ k = 2  4.61352 × 10⁻³−5.56763 × 10⁻⁴ −9.06592 × 10⁻⁶ −5.55578 × 10⁻⁸  4.86524 × 10⁻¹¹ 1.58714 × 10⁻¹² k = 4 −2.66903 × 10⁻⁶ −4.63010 × 10⁻⁸  3.94905 × 10⁻¹⁰ 1.39005 × 10⁻¹¹ −8.17065 × 10⁻¹⁴ −2.06989 × 10⁻¹⁵ k = 6  1.21750 × 10⁻⁹ 4.54096 × 10⁻¹¹  5.40143 × 10⁻¹³  2.52351 × 10⁻¹⁵  3.52067 × 10⁻¹⁸ k =8 −4.64097 × 10⁻¹⁴ −9.49010 × 10⁻¹⁶ −5.28855 × 10⁻¹⁸ k = 10  5.83933 ×10⁻¹⁹ Bjk j = 6 j = 7 j = 8 j = 9 j = 10 k = 0  1.44747 × 10⁻⁹  1.23245× 10⁻¹¹  2.11007 × 10⁻¹⁴ −1.81745 × 10⁻¹⁶ −6.94914 × 10⁻¹⁹ k = 2−1.17115 × 10⁻¹³ −1.25183 × 10⁻¹⁵ −3.66024 × 10⁻¹⁸ k = 4 −8.04925 ×10⁻¹⁸

[0236] TABLE 57 Example 7 Reflecting Surface S5 N0 = N1 = 1 LocalCoordinates x y z Position 83.9399 0.0000 0.0000 Vector VX 1.0000 0.00000.0000 VY 0.0000 1.0000 0.0000 VZ 0.0000 0.0000 1.0000 C0 0.000000

[0237] TABLE 58 Example 7 Projection Surface S6 N0 = N1 = 1 LocalCoordinates x y z Position −41.0601 −477.3845 0.0000 Vector VX −1.00000.0000 0.0000 VY 0.0000 −1.0000 0.0000 VZ 0.0000 0.0000 1.0000 C00.000000

[0238] TABLE 59 Example 8 Overall Size of Panel Display Surface I1 (mm):13.283 × 7.472 Size of Screen Surface I2 (mm): 1106 × 622 FNO inDirection of Longer Sides of Screen = 3.1 FNO in Direction of ShorterSides of Screen = 3.1

[0239] TABLE 60 Example 8 Surface Position and Rotation X Y Z X Y, ZSurface Coordinate Coordinate Coordinate Rotation Rotation Medium PanelDisplay Surface I1 0.000 680.462 −114.126 −24.938 0.000 Air First MirrorM1 Curved 0.000 665.860 −73.268 −11.118 0.000 Air Second Mirror M2Curved 0.000 682.460 −254.319 −12.824 0.000 Air Third Mirror M3 Curved0.000 616.446 −4.826 −10.840 0.000 Air Fourth Mirror M4 Curved 0.000341.827 −302.935 −55.262 0.000 Air Fifth Mirror M5 Flat 0.000 217.7468.580 8.596 0.000 Air Sixth Mirror M6 Flat 0.000 1.802 −300.000 0.0000.000 Air Screen Surface (I2) 0.000 0.000 0.000 0.000 0.000

[0240] TABLE 61 Example 8 Curved Surface Shape of First Mirror (M1)Radius of Curvature (mm) = 106377.218 K = −9.439 × 10⁴ C(0, 1) = 1.684 ×10⁻¹, C(2, 0) = −2.258 × 10⁻³, C(0,2) = −2.369 × 10⁻³ C(2, 1) = −9.083 ×10⁻⁶, C(0, 3) = 1.096 × 10⁻⁵, C(4,0) = 8.012 × 10⁻⁹ C(2, 2) = −4.285 ×10⁻⁸, C(0, 4) = 2.231 × 10⁻⁸

[0241] TABLE 62 Example 8 Curved Surface Shape of Second Mirror (M2)Radius of Curvature (mm) = 267.910 K = −4.304 × 10⁻¹ C(0, 1) = 4.674 ×10⁻², C(2, 0) = −1.645 × 10⁻⁴, C(0, 2)= −1.742 × 10⁻⁴ C(2, 1) = 6.569 ×10⁻⁸, C(0, 3) = 1.008 × 10⁻⁶, C(4, 0)= 7.508 × 10⁻¹⁰ C(2, 2) = −2.565 ×10⁻¹⁰, C(0, 4) = −2.336 × 10⁻⁹

[0242] TABLE 63 Example 8 Curved Surface Shape of Third Mirror (M3)Radius of Curvature (mm) = 570.148 K = −3.109 × 10² C(0, 1) = 7.535 ×10⁻², C(2, 0) = 5.066 × 10⁻⁴, C(0, 2) = 1.925 × 10⁻³ C(2, 1) = −1.293 ×10⁻⁵, C(0, 3) = −6.269 × 10⁻⁶, C(4, 0) = 2.246 × 10⁻⁷ C(2, 2) = 4.088 ×10⁻⁷, C(0, 4) = 5.380 × 10⁻⁸

[0243] TABLE 64 Example 8 Curved Surface Shape of Fourth Mirror (M4)Radius of Curvature (mm) = −993.286 K = 2.645 × 10 C(0, 1) = −1.238,C(2, 0) = −1.032 × 10⁻³, C(0, 2) = 1.167 × 10⁻³ C(2, 1) = 1.086 × 10⁻⁵,C(0, 3) = 6.456 × 10⁻⁶, C(4, 0) = 1.133 × 10⁻⁹ C(2, 2) = −1.795 × 10⁻⁷,C(0, 4) = −4.084 × 10⁻⁸

[0244] TABLE 65 Example 9 Overall Size of Panel Display Surface I1 (mm):   26.624 × 19.968 Size of Screen Surface I2 (mm):    1024 × 768 FNO inDirection of Longer Sides of Screen = 3.0 FNO in Direction of ShorterSides of Screen = 4.5

[0245] TABLE 66 Example 9 Surface Position and Rotation X Y Z X Y, ZSurface Coordinate Coordinate Coordinate Rotation Rotation Medium PanelDisplay Surface I1 0.000 1225.125 −127.470 −34.830 0.000 Air FirstMirror M1 Curved 0.000 1168.810 −111.456 −24.720 0.000 Air Aperture StopST 0.000 1235.920 −285.524 −25.369 0.000 Air Second Mirror M2 Curved0.000 1242.861 −287.234 −26.051 0.000 Air Third Mirror M3 Curved 0.0001136.690 −60.687 −24.388 0.000 Air Fourth Mirror M4 Curved 0.000 817.349−408.232 −81.766 0.000 Air Fifth Mirror M5 Flat 0.000 256.510 −42.517−2.937 0.000 Air Sixth Mirror M6 Flat 0.000 −5.416 −300.000 0.000 0.000Air Screen Surface (I2) 0.000 0.000 0.000 0.000 0.000

[0246] TABLE 67 Example 9 Curved Surface Shape of First Mirror (M1)Radius of Curvature (mm) = 47670.865 K = −6.434 × 10⁴ C(0, 1) = 2.113 ×10⁻¹, C(2, 0) = −9.706 × 10⁻⁴, C(0, 2) = −1.863 × 10⁻³ C(2, 1) = 7.277 ×10⁻⁶, C(0, 3) = 5.779 × 10⁻⁶, C(4, 0) = 1.856 × 10⁻⁸ C(2, 2) = 1.133 ×10⁻⁷, C(0, 4) = 9.931 × 10⁻⁸

[0247] TABLE 68 Example 9 Curved Surface Shape of Second Mirror (M2)Radius of Curvature (mm) = 266.862 K = −9.462 × 10⁻¹ C(0, 1) = −1.537 ×10⁻², C(2, 0) = −1.508 × 10⁻⁴, C(0, 2) = −1.841 × 10⁻⁴ C(2, 1) = 7.947 ×10⁻⁸, C(0, 3) = 2.548 × 10⁻⁷, C(4, 0) = 5.091 × 10⁻⁹ C(2, 2) = 9.101 ×10⁻⁹, C(0, 4) = 3.779 × 10⁻⁹

[0248] TABLE 69 Example 9 Curved Surface Shape of Third Mirror (M3)Radius of Curvature (mm) = 511.833 K = −1.186 × 10² C(0, 1) = 1.197 ×10⁻¹, C(2, 0) = 6.380 × 10⁻⁴, C(0, 2) = 1.251 × 10⁻³ C(2, 1) = −4.052 ×10⁻⁶, C(0, 3) = −3.980 × 10⁻⁶, C(4, 0) = 1.494 × 10⁻⁷ C(2, 2) = 2.385 ×10⁻⁷, C(0, 4) = 6.638 × 10⁻⁸

[0249] TABLE 70 Example 9 Curved Surface Shape of Fourth Mirror (M4)Radius of Curvature (mm) = 0.000 K = −1.879 × 10² C(0, 1) = −2.058, C(2,0) = −8.631 × 10⁻⁴, C(0, 2) = 1.908 × 10⁻³ C(2, 1) = 5.097 × 10⁻⁶, C(0,3) = 6.368 × 10⁻⁶, C(4, 0) = 2.201 × 10⁻⁹ C(2, 2) = −5.440 × 10⁻⁸, C(0,4) = −2.417 × 10⁻⁸

[0250] TABLE 71 Example 10 Overall Size of Panel Display Surface I1(mm):    13.283 × 7.472 Size of Screen Surface I2 (mm):    1106 × 622FNO in Direction of Longer Sides of Screen = 3.1 FNO in Direction ofShorter Sides of Screen = 3.1

[0251] TABLE 72 Example 10 Surface Position and Rotation X Y Z X Y, ZSurface Coordinate Coordinate Coordinate Rotation Rotation Medium PanelDisplay Surface I1 0.000 429.000 217.364 −72.761 0.000 Air First MirrorM1 Curved 0.000 379.122 202.375 −61.047 0.000 Air Second Mirror M2Curved 0.000 529.372 98.361 −62.009 0.000 Air Aperture Stop ST 0.000535.873 120.360 −63.273 0.000 Air Third Mirror M3 Curved 0.000 290.291211.413 −61.724 0.000 Air Fourth Mirror M4 Curved 0.000 297.822 −109.328−105.696 0.000 Air Fifth Mirror M5 Flat 0.000 −451.659 −322.158 −105.1090.000 Air Sixth Mirror M6 Flat 0.000 −715.000 250.000 0.000 0.000 AirScreen Surface (I2) 0.000 0.000 0.000 0.000 0.000

[0252] TABLE 73 Example 10 Curved Surface Shape of First Mirror (M1)Radius of Curvature (mm) = 55105.558 K = −9.416 × 10⁴ C(0, 1) = 2.665 ×10⁻¹, C(2, 0) = −1.488 × 10⁻³, C(0, 2) = −2.314 × 10⁻³ C(2, 1) = 7.735 ×10⁻⁶, C(0, 3) = 1.046 × 10⁻⁵, C(4, 0) = 8.701 × 10⁻⁹ C(2, 2) = 1.657 ×10⁻⁷, C(0, 4) = 8.682 × 10⁻⁸

[0253] TABLE 74 Example 10 Curved Surface Shape of Second Mirror (M2)Radius of Curvature (mm) = 266.893 K = 5.482 × 10⁻¹ C(0, 1) = 3.794 ×10⁻⁴, C(2, 0) = −1.355 × 10⁻⁴, C(0, 2) = −1.900 × 10⁻⁴ C(2, 1) = 3.553 ×10⁻⁷, C(0, 3) = 6.314 × 10⁻⁷, C(4, 0) = −5.225 × 10⁻⁹ C(2, 2) = −1.127 ×10⁻⁸, C(0, 4) = −6.737 × 10⁻⁹

[0254] TABLE 75 Example 10 Curved Surface Shape of Third Mirror (M3)Radius of Curvature (mm) = 669.724 K = −3.507 × 10² C(0, 1) = 9.594 ×10⁻², C(2, 0) = 9.897 × 10⁻⁴, C(0, 2) = 1.845 × 10⁻³ C(2, 1) = −3.058 ×10⁻⁶, C(0, 3) = −6.449 × 10⁻⁶, C(4, 0) = 2.237 × 10⁻⁷ C(2, 2) = 4.923 ×10⁻⁷, C(0, 4) = 1.248 × 10⁻⁷

[0255] TABLE 76 Example 10 Curved Surface Shape of Fourth Mirror (M4)Radius of Curvature (mm) = −1508.718 K = 6.332 × 10 C(0, 1) = −1.224,C(2, 0) = −4.739 × 10⁻⁴, C(0, 2) = 1.439 × 10⁻³ C(2, 1) = −3.909 × 10⁻⁷,C(0, 3) = 4.994 × 10⁻⁶, C(4, 0) = −2.171 × 10⁻⁸ C(2, 2) = −1.135 × 10⁻⁷,C(0, 4) = −2.627 × 10⁻⁸

[0256] TABLE 77 Examples 8 to 10 Values of Conditional Formulae, etc. DLHL DL/HL θ Example 8 1000 622 1.61 38.07 Example 9 1394 768 1.82 50.85Example 10 1650 622 2.65 55.68

1-10. (Cancelled).
 11. A rear projection optical system comprising aprojection optical system for projecting an image displayed on a displaysurface onto a screen surface, wherein the projection optical systemincludes at least four curved-surface reflecting mirrors, and whereinassuming that a ray of light that travels from a center of the paneldisplay surface to a center of the screen surface is called a screencenter ray, the following condition is fulfilled: 0.5<DL/HL<3.5,10<θ<70, Fnoy≦4.5, and Fnoz≦4.0, where: DL represents an opticaldistance traveled by the screen center ray from a last surface of theprojection optical system to the screen surface, HL represents adimension of the screen surface in a direction parallel to a planeformed at the center of the screen surface by a normal to the screensurface and the screen center ray, θ represents an angle of incidence atwhich the screen center ray is incident on the screen surface, Fnoyrepresents an f-number in a direction corresponding to the heightdirection of the display surface, and Fnoz represents an f-number in adirection corresponding to the width direction of the display surface.12. A rear projection optical system as claimed in claim 11, wherein thefollowing condition is fulfilled: Fnoy≧Fnoz.
 13. A rear projectionoptical system as claimed in claim 11, further comprising two flatsurface reflecting mirrors for turning an optical path of the projectionoptical system.
 14. A rear projection optical system as claimed in claim13, wherein the two flat surface reflecting mirrors are disposed on theoptical path between the screen surface and the curved reflectingmirrors.
 15. A rear projection optical system as claimed in claim 14,wherein one of the flat surface reflecting mirrors is parallel to thescreen.
 16. A rear projection optical system as claimed in claim 11,wherein, of the curved-surface reflecting mirrors, at least the oneclosest and the one second-closest to the display surface have asubstrate made of glass.
 17. A rear projection optical system as claimedin claim 11, wherein of the curved-surface reflecting mirrors, at leastthe one closest to the display surface has a substrate made of plastic18. A rear projection optical system as claimed in claim 11, furthercomprising the display panel for displaying the image on the displaysurface thereof, wherein focusing is achieved by moving the displaypanel along the screen center ray.
 19. A rear projection optical systemas claimed in claim 11, wherein of the curved-surface reflectingmirrors, focusing is achieved by moving the one closest or the onesecond closest to the display surface along the screen center ray.
 20. Arear projection optical system as claimed in claim 11, wherein zoomingis achieved by moving at least two curved-surface mirrors.
 21. Anoblique projection optical system for leading rays of light from adisplay surface on which an image is displayed to a projection surfacein such a way that a ray of light from a center of the display surfaceis obliquely incident on the projection surface in order to project amagnified image of the image displayed on the display surface onto theprojection surface, the oblique projection optical system comprising:includes at least four curved-surface reflecting mirrors, wherein thedisplay surface has a smaller dimension in a height direction than in awidth direction, the curved-surface reflecting mirrors reflecting therays of light from the display surface in such a way as to deflect therays of light in the height direction of the display surface, andwherein assuming that a ray of light that travels from a center of thepanel display surface to a center of the screen surface is called ascreen center ray, the following condition is fulfilled: 0.5<DL/HL<3.5,10<θ<70, Fnoy≦4.5, and Fnoz≦4.0, where: DL represents an opticaldistance traveled by the screen center ray from a last surface of theprojection optical system to the screen surface, HL represents adimension of the screen surface in a direction parallel to a planeformed at the center of the screen surface by a normal to the screensurface and the screen center ray, θ represents an angle of incidence atwhich the screen center ray is incident on the screen surface, Fnoyrepresents an f-number in a direction corresponding to the heightdirection of the display surface, and Fnoz represents an f-number in adirection corresponding to the width direction of the display surface.22. A rear projection optical system as claimed in claim 21, wherein thefollowing condition is fulfilled: Fnoy≧Fnoz.
 23. A rear projectionoptical system as claimed in claim 21, wherein, of the curved-surfacereflecting mirrors, at least the one closest and the one second-closestto the display surface have a substrate made of glass.
 24. A rearprojection optical system as claimed in claim 21, wherein of thecurved-surface reflecting mirrors, at least the one closest to thedisplay surface has a substrate made of plastic
 25. A rear projectionoptical system as claimed in claim 21, further comprising the displaypanel for displaying the image on the display surface thereof, whereinfocusing is achieved by moving the display panel along the screen centerray.
 26. A rear projection optical system as claimed in claim 21,wherein of the curved-surface reflecting mirrors, focusing is achievedby moving the one closest or the one second closest to the displaysurface along the screen center ray.
 27. A rear projection opticalsystem as claimed in claim 21, wherein zooming is achieved by moving atleast two curved-surface mirrors.